Research Article Quasi-Triangular Spaces, Pompeiu...

Research ArticleQuasi-Triangular Spaces Pompeiu-HausdorffQuasi-Distances and Periodic and Fixed Point Theorems ofBanach and Nadler Types

Kazimierz WBodarczyk

Department of Nonlinear Analysis Faculty of Mathematics and Computer Science University of Łodz Banacha 2290-238 Łodz Poland

Correspondence should be addressed to Kazimierz Włodarczyk wlkzxamathunilodzpl

Received 19 February 2015 Revised 17 May 2015 Accepted 18 May 2015

Academic Editor Poom Kumam

Copyright copy 2015 Kazimierz Włodarczyk This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Let119862 = 119862120572120572isinA isin [1infin)

AA-index set A quasi-triangular space (119883P119862A) is a set119883with familyP

119862A = 119901120572 119883

2

rarr [0infin) 120572 isin

A satisfying forall120572isinA forall

119906V119908isin119883 119901120572(119906 119908) le 119862

120572[119901

120572(119906 V) + 119901

120572(V 119908)] For any P

119862A a left (right) family J119862A generated by P

119862A isdefined to be J

119862A = 119869120572

1198832

rarr [0infin) 120572 isin A where forall120572isinA forall

119906V119908isin119883 119869120572(119906 119908) le 119862

120572[119869

120572(119906 V) + 119869

120572(V 119908)] and furthermore the

property forall120572isinA lim

119898rarrinfin119901120572(119908

119898 119906

119898) = 0 (forall

120572isinA lim119898rarrinfin

119901120572(119906

119898 119908

119898) = 0) holds whenever two sequences (119906

119898 119898 isin N) and

(119908119898

119898 isin N) in119883 satisfy forall120572isinA lim

119898rarrinfinsup

119899gt119898119869120572(119906

119898 119906

119899) = 0 and lim

119898rarrinfin119869120572(119908

119898 119906

119898) = 0 (forall


sup119899gt119898

119869120572(119906

119899 119906

119898) = 0

and lim119898rarrinfin

119869120572(119906

119898 119908

119898) = 0) In (119883P

119862A) using the left (right) familiesJ119862A generated byP

119862A (P119862A is a special case ofJ

119862A)we construct three types of Pompeiu-Hausdorff left (right) quasi-distances on 2

119883 for each typewe construct of left (right) set-valuedquasi-contraction 119879 119883 rarr 2

119883 and we prove the convergence existence and periodic point theorem for such quasi-contractionsWe also construct two types of left (right) single-valued quasi-contractions 119879 119883 rarr 119883 and we prove the convergence existenceapproximation uniqueness periodic point and fixed point theorem for such quasi-contractions (119883P

119862A) generalize ultra quasi-triangular and partiall quasi-triangular spaces (in particular generalize metric ultra metric quasi-metric ultra quasi-metric 119887-metric partial metric partial 119887-metric pseudometric quasi-pseudometric ultra quasi-pseudometric partial quasi-pseudometrictopological uniform quasi-uniform gauge ultra gauge partial gauge quasi-gauge ultra quasi-gauge and partial quasi-gaugespaces)

1 Introduction

The set-valued dynamic system is defined as a pair (119883 119879)where119883 is a certain space and119879 is a set-valuedmap119879 119883 rarr

2119883 here 2119883 denotes the family of all nonempty subsets of thespace119883 For119898 isin 0 cupN we define 119879

[119898]

= 119879 ∘ 119879 ∘ sdot sdot sdot ∘ 119879 (119898-times) and 119879

[0]= 119868

119883(an identity map on 119883) By Fix(119879) and

Per(119879)we denote the sets of all fixed points and periodic pointsof 119879 respectively that is Fix(119879) = 119908 isin 119883 119908 isin 119879(119908) andPer(119879) = 119908 isin 119883 119908 isin 119879

[119896]

(119908) for some 119896 isin N A dynamicprocess or a trajectory starting at 119908

0isin 119883 or a motion of the

system (119883 119879) at 1199080 is a sequence (119908119898

119898 isin 0 cup N) definedby 119908

119898

isin 119879(119908119898minus1

) for 119898 isin N (see [1ndash4])

Recall that a single-valued dynamic system is defined as apair (119883 119879) where119883 is a certain space and119879 is a single-valuedmap 119879 119883 rarr 119883 that is forall

119909isin119883119879(119909) isin 119883 By Fix(119879) and

Per(119879)we denote the sets of all fixed points and periodic pointsof 119879 respectively that is Fix(119879) = 119908 isin 119883 119908 = 119879(119908) andPer(119879) = 119908 isin 119883 119908 = 119879

[119896]

(119908) for some 119896 isin N For each119908

0isin 119883 a sequence (119908

119898

= 119879[119898]

(1199080) 119898 isin 0 cup N) is called

a Picard iteration starting at 1199080 of the system (119883 119879)Let 119883 be a (nonempty) set A distance on 119883 is a map 119901

1198832

rarr [0infin) The set 119883 together with distances on 119883 iscalled distance spaces

The following distance spaces are important for severalreasons

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2015 Article ID 201236 16 pageshttpdxdoiorg1011552015201236

2 Abstract and Applied Analysis

Definition 1 Let 119883 be a (nonempty) set and let 119901 1198832

rarr

[0infin)(A) (119883 119901) is calledmetric if (i)forall

119906119908isin119883119901(119906 119908) = 0 iff 119906=

119908 (ii) forall119906119908isin119883

119901(119906 119908) = 119901(119908 119906) and (iii) forall119906V119908isin119883 119901(119906

119908) le 119901(119906 V) + 119901(V 119908)(B) (See [5]) (119883 119901) is called ultra metric if (i) forall

119906119908isin119883

119901(119906 119908) = 0 iff 119906 = 119908 (ii) forall119906119908isin119883

119901(119906 119908) = 119901(119908 119906) and(iii) forall

119906V119908isin119883 119901(119906 119908) le max119901(119906 V) 119901(V 119908)(C) (See [6 7]) (119883 119901) is called 119887-metric with parameter

119862 isin [1infin) if (i) forall119906119908isin119883


119901(119906 119908) = 119901(119908 119906) and (iii) forall119906V119908isin119883 119901(119906 119908) le 119862[119901(119906 V) +

119901(V 119908)](D) (See [8]) (119883 119901) is called partial metric if (i) forall

119906119908isin119883

119906 = 119908 iff 119901(119906 119906) = 119901(119906 119908) = 119901(119908119908) (ii) forall119906119908isin119883

119901(119906

119906) le 119901(119906 119908) (iii) forall119906119908isin119883

119901(119906 119908) = 119901(119908 119906) and (iv)forall119906V119908isin119883 119901(119906 119908) le 119901(119906 V) + 119901(V 119908) minus 119901(V V)(E) (See [9]) (119883 119901) is called partial 119887-metric with param-

eter 119862 isin [1infin) if (i) forall119906119908isin119883

119906 = 119908 iff 119901(119906 119906) = 119901(119906

119908) = 119901(119908119908) (ii) forall119906119908isin119883

119901(119906 119906) le 119901(119906 119908) (iii) forall119906119908isin119883

119901(119906 119908) = 119901(119908 119906) and (iv) forall119906V119908isin119883 119901(119906 119908) le 119862[119901(119906 V) +

119901(V 119908)] minus 119901(V V)(F) (See [10]) (119883 119901) is called quasi-metric if (i) forall

119906119908isin119883

119901(119906 119908) = 0 iff 119906 = 119908 and (ii) forall119906V119908isin119883 119901(119906 119908) le 119901(119906 V) +

119901(V 119908)(G) (119883 119901) is called ultra quasi-metric if (i) forall

119906119908isin119883119901(119906

119908) = 0 iff 119906 = 119908 and (ii) forall119906V119908isin119883 119901(119906 119908) le max119901(119906 V)

119901(V 119908)(H) The distance 119901 is called pseudometric (or the gauge)

on119883 if (i) forall119906isin119883

119901(119906 119906) = 0 (ii) forall119906119908isin119883

119901(119906 119908) = 119901(119908 119906)and (iii) forall

119906V119908isin119883 119901(119906 119908) le 119901(119906 V) + 119901(V 119908)(I) The distance 119901 is called quasi-pseudometric (or the

quasi-gauge) on 119883 if (i) forall119906isin119883

119901(119906 119906) = 0 and (ii) forall119906V119908isin119883

119901(119906 119908) le 119901(119906 V) + 119901(V 119908)(J) (See [11]) The distance 119901 is called ultra quasi-

pseudometric (or the ultra quasi-gauge) on119883 if (i) forall119906isin119883

119901(119906119906) = 0 and (ii) forall

119906V119908isin119883 119901(119906 119908) le max119901(119906 V) 119901(V 119908)

Definition 2 (see [12]) Let 119883 be a (nonempty) set and letAbe an index set

(A) Each family D = 119889120572

120572 isin A of pseudometrics119889120572

1198832

rarr [0infin) 120572 isin A is called gauge on 119883 The gaugeD = 119889

120572 120572 isin A on 119883 is called separating if forall

119906119908isin119883119906 =

119908 rArr exist120572isinA 119889

120572(119906 119908) gt 0

(B) Let the familyD = 119889120572 120572 isin A be separating gauge

on 119883 The topology T(D) having as a subbase the familyB(D) = 119861(119906 119889

120572 120576

120572) 119906 isin 119883 120576

120572gt 0 120572 isin A of all balls

119861(119906 119889120572 120576

120572) = V isin 119883 119889

120572(119906 V) lt 120576

120572 with 119906 isin 119883 120576

120572gt 0 and

120572 isin A is called topology induced by D on 119883 the topologyT(D) is Hausdorff

(C) A topological space (119883T) such that there is aseparating gauge D on 119883 with T = T(D) is called a gaugespace and is denoted by (119883D)

Definition 3 (see [13]) Let 119883 be a (nonempty) set and let Abe an index set

(A) Each family P = 119901120572 120572 isin A of quasi-pseudom-

etrics 119901120572 119883

2rarr [0infin) 120572 isin A is called quasi-gauge on 119883

(B) Let the family P = 119901120572

120572 isin A be quasi-gaugeon 119883 The topology T(P) having as a subbase of the familyB(P) = 119861(119906 119901

120572 120576

120572) 119906 isin 119883 120576


119861(119906 119901120572 120576

120572) = V isin 119883 119901

120572(119906 V) lt 120576

120572 with 119906 isin 119883 120576

120572gt 0 and

120572 isin A is called topology induced byP on 119883(C) A topological space (119883T) such that there is a quasi-

gauge P on 119883 with T = T(P) is called quasi-gauge spaceand is denoted by (119883P)

Remark 4 (see [13 Theorems 42 and 26]) Each quasi-uniform space and each topological space is the quasi-gaugespace

There is a growing literature concerning set-valued andsingle-valued dynamic systems in the above defined distancespaces These studies contain also various extensions of theBanach [14] and Nadler [15 16] theorems Of course thereis a huge literature on this topic For some such spaces andtheorems in these spaces see for example M M Deza andE Deza [17] Kirk and Shahzad [18] and references therein

Recall that the first convergence existence approxima-tion uniqueness and fixed point result concerning single-valued contractions in complete metric spaces were obtainedby Banach in 1922 [14]

Theorem 5 (see [14]) Let (119883 119889) be a complete metric space If119879 119883 rarr 119883 and

exist0le120582lt1 forall119909119910isin119883

119889 (119879 (119909) 119879 (119910)) le 120582119889 (119909 119910) (1)

then the following are true (i)119879 has a unique fixed point119908 in119883

(ie there exists119908 isin 119883 such that119908 = 119879(119908) and Fix(119879) = 119908)and (ii) for each 119908

0isin 119883 the sequence (119879

[119898]

(1199080) 119898 isin N)

converges to 119908

The Pompeiu-Hausdorff metric 119867119889 on the class of all

nonempty closed and bounded subsetsCB(119883) of the metricspace (119883 119889) is defined as follows

119867119889

(119880119882) = maxsup119906isin119880

119889 (119906119882) sup119908isin119882

119889 (119908119880)

119880119882 isin CB (119883)

(2)

where for each 119909 isin 119883 and 119881 isin CB(119883) 119889(119909 119881) =

infVisin119881119889(119909 V) Using Pompeiu-Hausdorffmetric new contrac-tions were received by Nadler in 1967 and 1969 [15 16] as atool to study the existence of fixed points of set-valued mapsin complete metric spaces

Theorem 6 (see [15] [16 Theorem 5]) Let (119883 119889) be acomplete metric space If 119879 119883 rarr CB(119883) and

exist120582isin[01) forall

119909119910isin119883119867

119889

(119879 (119909) 119879 (119910)) le 120582119889 (119909 119910) (3)

then Fix(119879) = (ie there exists 119908 isin 119883 such that 119908 isin 119879(119908))

Markin [19 20] gave a slighty defferent version ofTheorem 6

Our primary interest is to construct new very general dis-tance spaces deliver new contractive set-valued and single-valued dynamic systems in these distance spaces present

Abstract and Applied Analysis 3

the new global methods for studying of these dynamicsystems in these spaces and prove new convergence approxi-mation existence uniqueness periodic point and fixed pointtheorems for such dynamic systems

The goal of the present paper is to introduce and describethe quasi-triangular spaces (119883P

119862A) (Section 2) and moregeneral quasi-triangular spaces (119883P

119862A) with left (right)families J

119862A generated by P119862A (Sections 3ndash5) Moreover

we use new methods and adopt ideas of Pompeiu andHausdorff (Section 7) (see [21] for an excellent introductionto these ideas) to establish in these spaces some versions ofBanach andNadler theorems (Sections 8 and 9) Here studieddynamic systems are left (right)J

119862A-admissible or left (right)P

119862A-closed (Section 6) Examples are provided (Sections 10ndash12) and concluding remarks are given (Section 13)

2 Quasi-Triangular Spaces (119883P119862A)

It is worth noticing that the distance spaces (119883P119862A) intro-

duced and described below are not necessarily topological orHausdorff or sequentially complete

Definition 7 Let119883 be a (nonempty) set letA be an index setand let 119862 = 119862

120572120572isinA isin [1infin)

A(A) One says that a family P

119862A = 119901120572

1198832

rarr

[0infin) 120572 isin A of distances is a quasi-triangular family on119883 if

forall120572isinA forall

119906V119908isin119883 119901120572(119906 119908) le119862

120572[119901

120572(119906 V) + 119901

120572(V 119908)] (4)

A quasi-triangular space (119883P119862A) is a set 119883 together with

the quasi-triangular familyP119862A on 119883

(B) Let (119883P119862A) be the quasi-triangular space One says

thatP119862A is separating if

forall119906119908isin119883

119906 =119908

997904rArrexist120572isinA 119901

120572(119906 119908) gt 0or119901

120572(119908 119906) gt 0

(5)

(C) If (119883P119862A) is an quasi-triangular space and forall

120572isinA

forall119906119908isin119883

119901minus1120572

(119906 119908) = 119901120572(119908 119906) then forall

120572isinA forall119906V119908isin119883 119901

minus1120572

(119906

119908) le 119862120572[119901

minus1120572

(119906 V) + 119901minus1120572

(V 119908)] One says that the quasi-triangular space (119883Pminus1

119862A) Pminus1119862A = 119901

minus1120572

1198832

rarr [0infin)

120572 isin A is the conjugation of (119883P119862A)

Remark 8 In the spaces (119883P119862A) in general the distan-

ces 119901120572

1198832

rarr [0infin) 120572 isin A do not vanish on the diago-nal they are asymmetric and do not satisfy triangle ine-quality (ie the properties forall

120572isinA forall119906isin119883

119901120572(119906 119906) = 0 or


119906119908isin119883119901

120572(119906 119908) = 119901

120572(119908 119906) or forall


120572(119906

119908) le 119901120572(119906 V) + 119901

120572(V 119908) do not necessarily hold) see

Section 10



A

(A) One says that a family L119862A = 119897

120572 119883

2rarr [0infin)

120572 isin A of distances on 119883 is a ultra quasi-triangular family if


119906V119908isin119883 119897120572(119906 119908)

le119862120572max 119897

120572(119906 V) 119897

120572(V 119908)

(6)

An ultra quasi-triangular space (119883L119862A) is a set 119883 together

with the ultra quasi-triangular familyL119862A on 119883

(B) One says that a family S119862A = 119904

120572 119883

2rarr [0infin)

120572 isin A of distances on 119883 is a partial quasi-triangular familyif


119906V119908isin119883 119904120572(119906 119908) le119862

120572[119904

120572(119906 V) + 119904

120572(V 119908)]

minus 119904120572(V V)

(7)

A partial quasi-triangular space (119883S119862A) is a set 119883 together

with the partial quasi-triangular family S119862A on 119883

Remark 10 It is worth noticing that quasi-triangular spacesgeneralize ultra quasi-triangular and partial quasi-triangularspaces (in particular generalize metric ultra metric quasi-metric ultra quasi-metric 119887-metric partial metric partial119887-metric pseudometric quasi-pseudometric ultra quasi-pseudometric partial quasi-pseudometric topological uni-form quasi-uniform gauge ultra gauge partial gauge quasi-gauge ultra quasi-gauge and partial quasi-gauge spaces)

3 Left (Right) Families J119862A Generated by

P119862A in Quasi-Triangular Spaces (119883P

119862A)

In themetric spaces (119883 119889) there are several types of distances(determined by 119889) which generalize metrics 119889 First thesedistances were introduced by Tataru [22] More generalconcepts of distances inmetric spaces (119883 119889)which generalize119889 of this sort are given by Kada et al [23] (119908-distances)Lin and Du [24] (120591-functions) Suzuki [25] (120591-distances)and Ume [26] (119906-distance) Distances in uniform spaceswere given by Valyi [27] In the appearing literature thesedistances and their generalizations in other spaces provideefficient tools to study various problems of fixed point theorysee for example [28ndash30] and references therein In this paperwe also generalize these ideas

LetP119862A be the quasi-triangular family on119883 It is natural

to define the notions of left (right) familiesJ119862A generated by

P119862A which provide new structures on 119883

Definition 11 Let (119883P119862A) be the quasi-triangular space

(A) The family J119862A = 119869

120572 120572 isin A of distances

119869120572

1198832

rarr [0infin) 120572 isin A is said to be a left (right) familygenerated byP

119862A if

(J1) forall120572isinA forall

119906V119908isin119883 119869120572(119906 119908) le 119862

120572[119869

120572(119906 V) + 119869

120572(V 119908)]

and furthermore


(J2) For any sequences (119906119898

119898 isin N) and (119908119898

119898 isin N) in119883 satisfying

forall120572isinA lim

119898rarrinfin

sup119899gt119898

119869120572(119906

119898 119906

119899) = 0 (8)

(forall120572isinA lim

119898rarrinfin

sup119899gt119898

119869120572(119906

119899 119906

119898) = 0) (9)


119898rarrinfin

119869120572(119908

119898 119906

119898) = 0 (10)


119898rarrinfin

119869120572(119906

119898 119908

119898) = 0) (11)

the following holds


119898rarrinfin

119901120572(119908

119898 119906

119898) = 0 (12)


119898rarrinfin

119901120572(119906

119898 119908

119898) = 0) (13)

(B) J119871(119883P119862A)

(J119877(119883P119862A)

) is the set of all left (right) familiesJ

119862A on 119883 generated byP119862A

Remark 12 From Definition 11 if follows that P119862A isin

J119871(119883P119862A)

cap J119877(119883P119862A)

Moreover there are families J119862A isin

J119871(119883P119862A)

andJ119862A isin J119877

(119883P119862A)such that the distances 119869

120572 120572 isin

A do not vanish on the diagonal are asymmetric and arequasi-triangular and thus are not metric ultra metric quasi-metric ultra quasi-metric 119887-metric partial metric partial119887-metric pseudometric (gauge) quasi-pseudometric (quasi-gauge) and ultra quasi-pseudometric (ultra quasi-gauge)

4 Relations between J119862A and P

119862A

Remark 13 The following result shows that Definition 11 iscorrect and that J119871

(119883P119862A) P

119862A = and J119877(119883P119862A)

P119862A =

Theorem 14 Let (119883P119862A) be the quasi-triangular space Let

119864 sub 119883 be a set containing at least two different points and let120583

120572120572isinA isin (0infin)

A where

forall120572isinA 120583

120572ge

120575120572(119864)

2119862120572


120572(119864) = sup 119901

120572(119906 119908) 119906 119908 isin119864

(14)

If J119862A = 119869

120572 120572 isin A where for each 120572 isin A the distance

119869120572 119883

2rarr [0infin) is defined by

119869120572(119906 119908) =

119901120572(119906 119908) 119894119891 119864 cap 119906 119908 = 119906 119908

120583120572

119894119891 119864 cap 119906 119908 = 119906 119908

(15)

thenJ119862A is left and right family generated byP

119862A

Proof Indeed we see that condition (J1) does not hold onlyif there exist some 1205720 isin A and 1199060 V0 1199080 isin 119883 such that

1198691205720

(1199060 1199080) gt 1198621205720

[1198691205720

(1199060 V0) + 1198691205720

(V0 1199080)] (16)

Then (15) implies 1199060 V0 1199080 cap 119864 = 1199060 V0 1199080 and thefollowing Cases 1ndash4 hold

Case 1 If 1199060 1199080 sub 119864 then V0 notin 119864 and by (16) and (15)1199011205720(1199060 1199080) gt 2119862

12057201205831205720 Therefore by (14) 119901

1205720(1199060 1199080) gt

211986212057201205831205720

ge 1205751205720(119864) This is impossible

Case 2 If 1199060 isin 119864 and 1199080 notin 119864 then (16) and (15) give 1205831205720

gt

1198621205720[119901

1205720(1199060 V0) + 120583

1205720] ge 119862

12057201205831205720

whenever V0 isin 119864 or 1205831205720

gt

1198621205720[120583

1205720+120583

1205720] = 2119862

12057201205831205720whenever V0 notin 119864This is impossible

Case 3 If 1199060 notin 119864 and 1199080 isin 119864 then (16) and (15) give 1205831205720

gt

1198621205720[120583

1205720+ 119901

1205720(V0 1199080)] ge 119862

12057201205831205720


gt

1198621205720[120583

1205720+120583

1205720] = 2119862


Case 4 If 1199060 notin 119864 and 1199080 notin 119864 then (16) and (15) give 1205831205720

gt

1198621205720[120583

1205720+ 120583

1205720] = 2119862

12057201205831205720for V0 isin 119883 This is impossible

Therefore forall120572isinA forall

119906V119908isin119883 119869120572(119906 119908) le 119862

120572[119869

120572(119906 V) + 119869

120572(V

119908)] that is the condition (J1) holdsAssume now that the sequences (119906

119898 119898 isin N) and (119908

119898

119898 isin N) in119883 satisfy (8) and (10)Then (12) holds Indeed (10)implies

forall120572isinA forall0lt120576lt120583120572 exist

1198980=1198980(120572)isinNforall119898ge1198980

119869120572(119908

119898 119906

119898) lt 120576 (17)

Denoting 1198981015840

= min1198980(120572) 120572 isin A we see by(17) and (15) that forall

119898ge119898

1015840 119864 cap 119908119898 119906

119898 = 119908

119898 119906

119898

Then in view of Definition 11(A) (15) and (17) this impliesforall120572isinA forall0lt120576lt120583120572 exist

1198981015840isinN forall

119898ge1198981015840 119901

120572(119908

119898 119906

119898) = 119869

120572(119908

119898 119906

119898) lt 120576

Hence we obtain that the sequences (119906119898

119898 isin N) and(119908

119898 119898 isin N) satisfy (12) Thus we see thatJ

119862A is left familygenerated byP

119862AIn a similar way we show that (13) holds if (119906

119898 119898 isin N)

and (119908119898

119898 isin N) in 119883 satisfy (9) and (11) Therefore J119862A

is right family generated by P119862A We proved that J

119862A isin

J119871(119883P119862A)

cap J119877(119883P119862A)

holds

The following is interesting in respect to its use

Theorem 15 Let (119883P119862A) be the quasi-triangular space and

let J119862A be the left (right) family generated by P

119862A If P119862A

is separating on 119883 (ie (5) holds) then J119862A is separating on

119883 that is

forall119906119908isin119883

119906 =119908


120572(119906 119908) gt 0or 119869

120572(119908 119906) gt 0

(18)

holds

Proof We begin by supposing that 1199060 1199080 isin 119883 1199060 = 1199080 andforall120572isinA 119869

120572(1199060 1199080) = 0 and 119869

120572(1199080 1199060) = 0 Then (J1) implies


120572(1199060 1199060) le 119862

120572[119869

120572(1199060 1199080)+119869

120572(1199080 1199060)] = 0 or equiv-

alently forall120572isinA 119869

120572(1199060 1199060) = 119869

120572(1199080 1199060) = 0 and forall

120572isinA 119869120572(1199060

1199060) = 119869120572(1199060 1199080) = 0 Assuming that 119906

119898= 1199060 and 119908

119898= 1199080

119898 isin N we conclude that forall120572isinA lim

119898rarrinfinsup

119899gt119898119869120572(119906

119898 119906

119899) =

lim119898rarrinfin

119869120572(119908

119898 119906

119898) = 0 and forall


sup119899gt119898

119869120572(119906

119899

119906119898) = lim

119898rarrinfin119869120572(119906

119898 119908

119898) = 0 Therefore it is not

hard to see that (8)ndash(11) hold and by (J2) the above


considerations lead to the following conclusion 1199060 = 1199080 and


119898rarrinfin119901120572(119908

119898 119906

119898) = lim

119898rarrinfin119901120572(119906

119898 119908

119898) = 0 or

equivalently 1199060 = 1199080 and forall120572isinA 119901

120572(1199080 1199060) = 119901

120572(1199060 1199080) =

0 HoweverP119862A is separating A contradiction Therefore

J119862A is separating

5 Left (Right) J119862A-Convergences and Left

(Right) J119862A-Sequentially Completeness


and letJ119862A be the left (right) family generated byP

119862A(A) One says that a sequence (119906

119898 119898 isin N) sub 119883 is left

(right)J119862A-Cauchy sequence ifforall120572isinA lim

119898rarrinfinsup

119899gt119898119869120572(119906

119898

119906119899) = 0 (forall


sup119899gt119898

119869120572(119906

119899 119906

119898) = 0)

(B) Let 119906 isin 119883 and let (119906119898

119898 isin N) sub 119883 One says thatthe sequence (119906

119898 119898 isin N) is left (right) J

119862A-convergent to119906 if 119906 isin LIM119871minusJ119862A

(119906119898 119898isinN)= (119906 isin LIM119877minusJ119862A

(119906119898 119898isinN)= ) where

LIM119871minusJ119862A(119906119898 119898isinN)

= 119909

isin119883 forall120572isinA lim

119898rarrinfin

119869120572(119909 119906

119898) = 0

(LIM119877minusJ119862A(119906119898 119898isinN)

= 119909 isin119883 forall120572isinA lim

119898rarrinfin

119869120572(119906

119898 119909) = 0)

(19)

(C) One says that a sequence (119906119898

119898 isin N) sub

119883 is left (right) J119862A-convergent in 119883 if LIM119871minusJ119862A

(119906119898 119898isinN)=


= )(D) If every left (right) J

119862A-Cauchy sequence (119906119898

119898 isin N) sub 119883 is left (right) J119862A-convergent in 119883 (ie


= (LIM119877minusJ119862A(119906119898 119898isinN)

= )) then (119883P119862A) is

called left (right)J119862A-sequentially complete

Remark 17 The structures on 119883 determined by left (right)families J

119862A generated by P119862A are more general than the

structure on 119883 determined byP119862A see Remark 34

Remark 18 Let (119883P119862A) be the quasi-triangular space It is

clear that if (119906119898

119898 isin N) is left (right) P119862A-convergent

in 119883 then LIM119871minusP119862A(119906119898 119898isinN)

sub LIM119871minusP119862A(V119898 119898isinN)

(LIM119877minusP119862A(119906119898 119898isinN)

sub

LIM119877minusP119862A(V119898 119898isinN)

) for each subsequence (V119898

119898 isin N) of (119906119898

119898 isin N)

Definition 19 One says that (119883P119862A) is left (right)Hausdorff

if for each left (right) P119862A-convergent in 119883 sequence (119906

119898

119898 isin N) the set LIM119871minusP119862A(119906119898 119898isinN)


) is a singleton

6 Left (Right) J119862A-Admissible and Left

(Right) P119862A-Closed Set-Valued Maps

The following terminologies will be much used in the sequel



119862A Let(119883 119879) be the set-valued dynamic system 119879 119883 rarr 2119883

(A) Given 1199080

isin 119883 One says that (119883 119879) is left (right)J

119862A-admissible in 1199080 if for each dynamic processes (119908

119898

119898 isin 0 cup N) starting at 1199080 forall

119898isin0cupN 119908119898+1

isin 119879(119908119898

)LIM119871minusJ119862A

(119908119898119898isin0cupN) = (LIM119877minusJ119862A

(119908119898119898isin0cupN) = ) whenever


119898rarrinfin

sup119899gt119898

119869120572(119908

119898

119908119899

) = 0


119898rarrinfin

sup119899gt119898

119869120572(119908

119899

119908119898

) = 0)

(20)

(B) One says that (119883 119879) is left (right)J119862A-admissible on

119883 if (119883 119879) is left (right) J119862A-admissible in each point 1199080

isin

119883

Remark 21 Let (119883P119862A) be the quasi-triangular space and


119862A Let(119883 119879) be the set-valued dynamic system on 119883 If (119883P

119862A)

is left (right) J119862A-sequentially complete then (119883 119879) is left

(right)J119862A-admissible on119883 but the converse not necessarily

holds

We can define also the following generalization of conti-nuity


Let (119883 119879) be the set-valued dynamic system 119879 119883 rarr 2119883and let 119896 isin NThe set-valued dynamic system (119883 119879

[119896]

) is saidto be a left (right) P

119862A-closed on 119883 if for every sequence(119909

119898 119898 isin N) in 119879

[119896]

(119883) left (right) P119862A-converging in

119883 (thus LIM119871minusP119862A(119909119898 119898isinN)

= (LIM119877minusP119862A(119909119898 119898isinN)

= )) and havingsubsequences (V

119898 119898 isin N) and (119906

119898 119898 isin N) satisfying

forall119898isinN V

119898isin 119879

[119896]

(119906119898) the following property holds there

exists 119909 isin LIM119871minusP119862A(119909119898 119898isinN)

(119909 isin LIM119877minusP119862A(119909119898 119898isinN)

) such that 119909 isin

119879[119896]

(119909) (119909 isin 119879[119896]

(119909))

7 Left (Right) Pompeiu-HausdorffQuasi-Distances and Left (Right)Set-Valued Quasi-Contractions

In this section in the quasi-triangular spaces (119883P119862A)

using left (right) familiesJ119862A generated byP

119862A we definethree types of left (right) Pompeiu-Hausdorff quasi-distanceson 2119883 and for each type a left (right) set-valued quasi-contraction 119879 119883 rarr 2119883 is constructed



119862A Let120582 = 120582

120572120572isinA isin [0 1)A let (119883 119879) be a set-valued dynamic

system 119879 119883 rarr 2119883 and let 120578 isin 1 2 3 Let


119909isin119883forall119881isin2119883 119869

120572(119909 119881) = inf 119869

120572(119909 V) V isin119881

and 119869120572(119881 119909) = inf 119869

120572(V 119909) V isin119881

(21)


(A) LetJ119862A isin J119871

(119883P119862A) If


119880119882isin2119883 119863119871minusJ119862A

12119883120572 (119880119882)

=maxsup119906isin119880

119869120572(119906119882) sup

119908isin119882

119869120572(119880 119908)


119880119882isin2119883 119863119871minusJ119862A

22119883120572 (119880119882)


119869120572(119906119882) sup

119908isin119882

119869120572(119908 119880)


119880119882isin2119883 119863119871minusJ119862A

32119883120572 (119880119882) = sup119906isin119880

119869120572(119906119882)

(22)

then a family D119871minusJ119862A

1205782119883 = 119863119871minusJ119862A

1205782119883120572 120572 isin A is said to be left

D119871minusJ119862A

1205782119883 -quasi-distance on 2119883If


119909119910isin119883119862

120572sdot 119863

119871minusJ119862A

1205782119883120572 (119879 (119909) 119879 (119910))

le 120582120572119869120572(119909 119910)

(23)

thenwe say that (119883 119879) is a left (D119871minusJ119862A

1205782119883 120582)-quasi-119888119900119899119905119903119886119888119905119894119900119899on 119883

(B) LetJ119862A isin J119877

(119883P119862A) If


119880119882isin2119883 119863119877minusJ119862A

12119883120572 (119880119882)


119869120572(119906119882) sup

119908isin119882

119869120572(119880 119908)


119880119882isin2119883 119863119877minusJ119862A

22119883120572 (119880119882)


119869120572(119906119882) sup

119908isin119882

119869120572(119908 119880)


119880119882isin2119883 119863119877minusJ119862A

32119883120572 (119880119882) = sup119906isin119880

119869120572(119906119882)

(24)

then a familyD119877minusJ119862A

1205782119883 = 119863119877minusJ119862A

1205782119883120572 120572 isin A is said to be right




119909119910isin119883119862

120572sdot 119863

119877minusJ119862A

1205782119883120572 (119879 (119909) 119879 (119910))

le 120582120572119869120572(119909 119910)

(25)

then we say that (119883 119879) is a right (D119877minusJ119862A

1205782119883 120582)-quasi-contrac-tion on 119883

Remark 24 Observe that D119871minusJ119862A

1205782119883 and D119877minusJ119862A

1205782119883 extend (2)Quasi-contractions (23) and (25) extend (3)

Remark 25 Each (D119871minusJ119862A

1205782119883 120582)-quasi-contraction ((D119877minusJ119862A

1205782119883

120582)-quasi-contraction) 120578 isin 1 2 is (D119871minusJ119862A

32119883 120582)-quasi-con-traction ((D

119877minusJ119862A

32119883 120582)-quasi-contraction) but the conversedoes not necessarily hold

8 Convergence Existence Approximationand Periodic Point Theorem ofNadler Type for Left (Right) Set-ValuedQuasi-Contractions

The following result extends Theorem 6 to spaces (119883P119862A)

Theorem26 Let (119883P119862A) be the quasi-triangular space and

let (119883 119879) be the set-valued dynamic system 119879 119883 rarr 2119883 Let120578 isin 1 2 3 and let 120582 = 120582

120572120572isinA isin [0 1)A

Assume that there exist a left (right) familyJ119862A generated

byP119862A and a point 1199080

isin 119883 with the following properties(A1) (119883 119879) is left (D

119871minusJ119862A

1205782119883 120582)-quasi-contraction (right

(D119877minusJ119862A

1205782119883 120582)-quasi-contraction) on 119883

(A2) (119883 119879) is left (right)J119862A-admissible in 119908

0(A3) For every119909 isin 119883 and for every120573 = 120573


A

there exists 119910 isin 119879(119909) such that


120572(119909 119910) lt 119869

120572(119909 119879 (119909)) + 120573

120572 (26)

(forall120572isinA 119869

120572(119910 119909) lt 119869

120572(119879 (119909) 119909) + 120573

120572) (27)

Then the following hold(B1) There exist a dynamic process (119908

119898

119898 isin 0 cup N)

of the system (119883 119879) starting at 1199080 forall119898isin0cupN 119908

119898+1isin 119879(119908

119898

)and a point 119908 isin 119883 such that (119908119898

119898 isin 0 cup N) is left (right)P

119862A-convergent to 119908(B2) If the set-valued dynamic system (119883 119879

[119896]

) is left (right)P

119862A-closed on 119883 for some 119896 isin N then Fix(119879[119896]

) = andthere exist a dynamic process (119908119898

119898 isin 0 cupN) of the system(119883 119879) starting at 1199080 forall

119898isin0cupN 119908119898+1

isin 119879(119908119898

) and a point119908 isin Fix(119879[119896]

) such that (119908119898


119862A-convergent to 119908

Proof We prove only the case when J119862A is a left family

generated by P119862A (119883 119879) is left J

119862A-admissible in a point119908

0isin 119883 and (119883 119879

[119896]

) is left P119862A-closed on 119883 The case of

ldquorightrdquo will be omitted since the reasoning is based on theanalogous technique

Part 1 Assume that (A1)ndash(A3) holdBy (21) and the fact that 119869

120572 119883

2rarr [0infin) 120572 isin A we

choose

119903 = 119903120572120572isinA

isin (0infin)A (28)


such that


120572(119908

0 119879 (119908

0)) lt(1minus

120582120572

119862120572

) 119903120572 (29)

Put


(0)120572

=(1minus120582120572

119862120572

) 119903120572minus 119869

120572(119908

0 119879 (119908

0)) (30)

In view of (28) and (29) this implies 120573(0)

= 120573(0)120572

120572isinA isin

(0infin)A and we apply (26) to find 119908

1isin 119879(119908

0) such that


120572(119908

0 119908

1) lt 119869

120572(119908

0 119879 (119908

0)) + 120573

(0)120572

(31)

We see from (30) and (31) that


120572(119908

0 119908

1) lt(1minus

120582120572

119862120572

) 119903120572 (32)

Put now


(1)120572

=(120582120572

119862120572

)[(1minus120582120572

119862120572

) 119903120572minus 119869

120572(119908

0 119908

1)] (33)

Then in view of (32) we get 120573(1)= 120573

(1)120572


A andapplying again (26) we find 119908

2isin 119879(119908

1) such that


120572(119908

1 119908

2) lt 119869

120572(119908

1 119879 (119908

1)) + 120573

(1)120572

(34)

Observe that


120572(119908

1 119908

2) lt(

120582120572

119862120572

)(1minus120582120572

119862120572

) 119903120572 (35)

Indeed from (34) Definition 23(A) and using (33) in theevent that 120578 = 1 or 120578 = 2 or 120578 = 3 we get


120572(119908

1 119908

2) lt 119869

120572(119908

1 119879 (119908

1)) + 120573

(1)120572

le sup 119869120572(119906 119879 (119908

1)) 119906 isin119879 (119908

0) + 120573

(1)120572

le119863119871minusJ119862A

1205782119883120572 (119879 (1199080) 119879 (119908

1)) + 120573

(1)120572

le(120582120572

119862120572

) 119869120572(119908

0 119908

1) + 120573

(1)120572

=(120582120572

119862120572

)(1minus120582120572

119862120572

) 119903120572

(36)

Thus (35) holdsNext define


(2)120572

=(120582120572

119862120572

)[(120582120572

119862120572

)(1minus120582120572

119862120572

) 119903120572minus 119869

120572(119908

1 119908

2)]

(37)

Then in view of (35) 120573(2)= 120573

(2)120572


A Applying(26) in this situation we conclude that there exists 119908

3isin

119879(1199082) such that


120572(119908

2 119908

3) lt 119869

120572(119908

2 119879 (119908

2)) + 120573

(2)120572

(38)

We seek to show that


120572(119908

2 119908

3) lt(

120582120572

119862120572

)

2

(1minus120582120572

119862120572

) 119903120572 (39)

By (38) Definition 23(A) and using (37) in the event that120578 = 1 or 120578 = 2 or 120578 = 3 it follows that


120572(119908

2 119908

3) lt 119869

120572(119908

2 119879 (119908

2)) + 120573

(2)120572

⩽ sup119906isin119879(119908

1)

119869120572(119906 119879 (119908

2)) + 120573

(2)120572

le119863119871minusJ119862A

1205782119883120572 (119879 (1199081) 119879 (119908

2)) + 120573

(2)120572

le(120582120572

119862120572

) 119869120572(119908

1 119908

2) + 120573

(2)120572

=(120582120572

119862120572

)

2

(1minus120582120572

119862120572

) 119903120572

(40)

Thus (39) holdsProceeding as before using Definition 23(A) we get that

there exists a sequence (119908119898

119898 isin N) in 119883 satisfying

forall119898isinN 119908

119898+1isin119879 (119908

119898

) (41)

and for calculational purposes upon letting forall119898isinN 120573

(119898)

=

120573(119898)

120572120572isinA

where


119898isinN 120573(119898)

120572=(

120582120572

119862120572

)

sdot [(120582120572

119862120572

)

119898minus1(1minus

120582120572

119862120572

) 119903120572minus 119869

120572(119908

119898minus1 119908

119898

)]

(42)

we observe that forall119898isinN 120573

(119898)

isin (0infin)A


119898isinN 119869120572(119908

119898

119908119898+1

) lt 119869120572(119908

119898

119879 (119908119898

))

+ 120573(119898)

120572

(43)


119898isinN 119869120572(119908

119898

119908119898+1

)

lt(120582120572

119862120572

)

119898

(1minus120582120572

119862120572

) 119903120572

(44)


Let now 119898 lt 119899 Using (J1) we get

forall120572isinA

119869120572(119908

119898

119908119899

) ⩽119862120572119869120572(119908

119898

119908119898+1

)

+1198622120572119869120572(119908

119898+1 119908

119898+2) + sdot sdot sdot

+ 119862119899minus119898minus1120572

119869120572(119908

119899minus2 119908

119899minus1) +119862

119899minus119898minus1120572

119869120572(119908

119899minus1 119908

119899

)

=

119899minus119898minus2sum

119895=0119862119895+1120572

119869120572(119908

119898+119895

119908119898+119895+1

)

+119862119899minus119898minus1120572

119869120572(119908

119899minus1 119908

119899

)

(45)

Hence by (44) for each 120572 isin A

119869120572(119908

119898

119908119899

) lt (1minus120582120572

119862120572

)

sdot 119903120572

[

[


119895=0119862119895+1120572

(120582120572

119862120572

)

119898+119895

+119862119899minus119898minus1120572

(120582120572

119862120572

)

119899minus2]

]

= (1minus120582120572

119862120572

)

sdot 119903120572

[

[

119862120572(

120582120572

119862120572

)

119898 119899minus119898minus2sum

119895=0120582119895

120572+(

119862120572

1205822120572

)120582119899

120572

119862119898

120572

]

]

(46)

This and (41) mean that

exist(119908119898119898isinN) forall

119898isin0cupN 119908119898+1

isin119879 (119908119898

) (47)

and since 119898 lt 119899 implies 120582119899

120572le 120582

119898

120572


119898rarrinfin

sup119899gt119898

119869120572(119908

119898

119908119899

) le lim119898rarrinfin

sup119899gt119898

(1minus120582120572

119862120572

)

sdot 119903120572[119862

120572(

120582120572

119862120572

)

119898

(1 minus 120582120572)minus1

+(119862120572

1205822120572

)120582119899

120572

119862119898

120572

]

le lim119898rarrinfin

(1minus120582120572

119862120572

) 119903120572[119862

120572(

120582120572

119862120572

)

119898

(1minus120582120572)minus1

+(119862120572

1205822120572

)(120582120572

119862120572

)

119898

]= 0

(48)

Now since (119883 119879) is left J119862A-admissible in 119908

0isin 119883 by

Definition 20(A) properties (47) and (48) imply that thereexists 119908 isin 119883 such that


119898rarrinfin

119869120572(119908 119908

119898

) = 0 (49)

Next defining 119906119898

= 119908119898 and 119908

119898= 119908 for 119898 isin N by

(48) and (49) we see that conditions (8) and (10) hold for

the sequences (119906119898

119898 isin N) and (119908119898

119898 isin N) in 119883Consequently by (J2) we get (12) which implies that


119898rarrinfin

119901120572(119908 119908

119898

) = lim119898rarrinfin

119901120572(119908

119898 119906

119898) = 0 (50)

and so in particular we see that 119908 isin LIM119871minusP119862A(119908119898119898isinN)

Part 2 Assume that (A1)ndash(A3) hold and that for some 119896 isin N(119883 119879

[119896]

) is left P119862A-closed on 119883

By Part 1 LIM119871minusP119862A(119908119898119898isin0cupN) = and since by (47)

119908(119898+1)119896

isin 119879[119896]

(119908119898119896

) for 119898 isin 0 cup N thus defining (119909119898

=

119908119898minus1+119896

119898 isin N) we see that (119909119898

119898 isin N) sub 119879[119896]

(119883)LIM119871minusP119862A

(119909119898 119898isin0cupN) = LIM119871minusP119862A(119908119898119898isin0cupN) = the sequences (V

119898=

119908(119898+1)119896

119898 isin N) sub 119879[119896]

(119883) and (119906119898

= 119908119898119896

119898 isin N) sub

119879[119896]

(119883) satisfy forall119898isinN V

119898isin 119879

[119896]

(119906119898) and as subsequences

of (119909119898

119898 isin 0 cup N) are left P119862A-converging to each

point of the set LIM119871minusP119862A(119908119898119898isin0cupN) Moreover by Remark 18

LIM119871minusP119862A(119908119898119898isinN)


and LIM119871minusP119862A(119908119898119898isinN)

sub LIM119871minusP119862A(119906119898 119898isinN)

By the above and by Definition 22 since 119879

[119896] is left P119862A-

closed we conclude that there exist 119908 isin LIM119871minusP119862A(119908119898119898isin0cupN) =

LIM119871minusP119862A(119909119898 119898isinN)

such that 119908 isin 119879[119896]

(119908)

Part 3 The result now follows at once from Parts 1 and 2

9 Theorem of Banach Type in Quasi-Triangular Spaces (119883P

119862A)


using left (right) families J119862A generated by P

119862A weconstruct two types of left (right) single-valued quasi-contractions 119879 119883 rarr 119883 and convergence existenceapproximation uniqueness periodic point and fixed pointtheorem for such quasi-contractions is also proved

The following Definition 27 can be stated as a single-valued version of Definition 23


and let J119862A be the left (right) family generated by P

119862ALet (119883 119879) be the single-valued dynamic system let 120582 =

120582120572120572isinA isin [0 1)A and let 120578 isin 1 2(A) If J

119862A isin J119871(119883P119862A)

then we define the left D119871minusJ119862A119883120578

-

quasi-distance on 119883 byD119871minusJ119862A119883120578

= 119863119871minusJ119862A120578119883120572

1198832

rarr [0infin)120572 isin A where


119906119908isin119883119863

119871minusJ119862A1119883120572 (119906 119908)

=max 119869120572(119906 119908) 119869

120572(119908 119906)


119906119908isin119883119863

119871minusJ119862A2119883120572 (119906 119908) = 119869

120572(119906 119908)

(51)


One says that (119883 119879) is left (D119871minusJ119862A119883120578

120582)-quasi-contraction on119883 if


119909119910isin119883119862

120572sdot 119863

119871minusJ119862A120578119883120572

(119879 (119909) 119879 (119910))

le 120582120572119869120572(119909 119910)

(52)

(B) If J119862A isin J119877

(119883P119862A) then one defines the right

D119877minusJ119862A119883120578

-quasi-distance on119883 byD119877minusJ119862A119883120578

= 119863119877minusJ119862A120578119883120572

1198832

rarr

[0infin) 120572 isin A where


119906119908isin119883119863

119877minusJ119862A1119883120572 (119906 119908)

=max 119869120572(119906 119908) 119869

120572(119908 119906)


119906119908isin119883119863

119877minusJ119862A2119883120572 (119906 119908) = 119869

120572(119906 119908)

(53)

One says that (119883 119879) is right (D119877minusJ119862A119883120578

120582)-quasi-contractionon 119883 if


119909119910isin119883119862

120572sdot 119863

119877minusJ119862A120578119883120572

(119879 (119909) 119879 (119910))

le 120582120572119869120572(119909 119910)

(54)

Remark 28 Observe that (52) and (54) extend (1)

The following terminologies will be much used in thesequel



119862A Let(119883 119879) be the single-valued dynamic system 119879 119883 rarr 119883

(A) Given 1199080


119862A-admissible in 1199080 if for the sequence (119908

119898

= 119879[119898]

(1199080)

119898 isin 0 cup N) LIM119871minusJ119862A(119908119898119898isin0cupN) = (LIM119877minusJ119862A

(119908119898119898isin0cupN) = )

whenever


119898rarrinfin

sup119899gt119898

119869120572(119908

119898

119908119899

) = 0


119898rarrinfin

sup119899gt119898

119869120572(119908

119899

119908119898

) = 0)

(55)

(B)We say that (119883 119879) is left (right)J119862A-admissible on119883

if (119883 119879) is left (right)J119862A-admissible in each point 1199080

isin 119883

Remark 30 Let (119883P119862A) be the quasi-triangular space


119862ALet (119883 119879) be the single-valued dynamic system on 119883 If(119883P

119862A) is left (right) J119862A-sequentially complete then

(119883 119879) is left (right)J119862A-admissible on 119883

We can define the following generalization of continuity


Let (119883 119879) be the single-valued dynamic system 119879 119883 rarr 119883and let 119896 isin N The single-valued dynamic system (119883 119879

[119896]

) is

said to be a left (right)P119862A-closed on 119883 if for each sequence

(119909119898

119898 isin N) in 119879[119896]





119898 119898 isin N) and (119906


forall119898isinN V

119898= 119879

[119896]




) such that 119909 =

119879[119896]

(119909) (119909 = 119879[119896]

(119909))

The following result extends Theorem 5 to spaces (119883P

119862A)


let (119883 119879) be the single-valued dynamic system 119879 119883 rarr 2119883Let 120578 isin 1 2 and let 120582 = 120582

120572120572isinA isin [0 1)A




119871minusJ119862A119883120578

120582)-quasi-contraction (right(D

119877minusJ119862A119883120578

120582)-quasi-contraction) on 119883(A2) (119883 119879) is left (right) J

119862A-admissible in a point 1199080isin

119883Then the following hold(B1) There exists a point 119908 isin 119883 such that the sequence

(119908119898

= 119879[119898]

(1199080) 119898 isin 0 cup N) starting at 1199080 is left (right)

P119862A-convergent to 119908(B2) If the single-valued dynamic system (119883 119879

[119896]

) is left(right)P


) = there exists a point 119908 isin Fix(119879[119896]

) such that the sequence(119908

119898

= 119879[119898]


P119862A-convergent to 119908 and

forall120572isinA forallVisinFix(119879[119896]) 119869

120572(V 119879 (V)) = 119869

120572(119879 (V) V) = 0 (56)

(B3) If the familyP119862A = 119901

120572 120572 isin A is separating on 119883

and if the single-valued dynamic system (119883 119879[119896]

) is left (right)P

119862A-closed on 119883 for some 119896 isin N then there exists a point119908 isin 119883 such that

Fix (119879[119896]

) = Fix (119879) = 119908 (57)

the sequence (119908119898

= 119879[119898]

(1199080) 119898 isin 0 cup N) starting at 1199080 is

left (right)P119862A- convergent to 119908 and


120572(119908 119908) = 0 (58)

Proof By Theorem 26 we prove only (56)ndash(58) and only inthe case of ldquoleftrdquo We omit the proof in the case of ldquorightrdquowhich is based on the analogous technique

Part 1Property (56) holdsSuppose thatexist1205720isinA

existVisinFix(119879[119896]) 1198691205720(V119879(V)) gt 0 Of course V = 119879

[119896]

(V) = 119879[2119896]

(V) 119879(V) =

119879[2119896]

(119879(V)) and for 120578 isin 1 2 by Definition 27(A)

0 lt 1198691205720

(V 119879 (V)) = 1198691205720

(119879[2119896]

(V) 119879[2119896](119879 (V)))

le 119863119871minusJ119862A1205781198831205720

(119879[2119896]

(V) 119879[2119896](119879 (V)))


le (1205821205720

1198621205720

)1198691205720

(119879[2119896minus1]

(V) 119879[2119896minus1](119879 (V)))

le (1205821205720

1198621205720

) sdot119863119871minusJ119862A1205781198831205720

(119879[2119896minus1]

(V) 119879[2119896minus1](119879 (V)))

le (1205821205720

1198621205720

)

2

1198691205720

(119879[2119896minus2]

(V) 119879[2119896minus2](119879 (V))) le sdot sdot sdot

le (1205821205720

1198621205720

)

2119896

1198691205720

(V 119879 (V)) lt 1198691205720

(V 119879 (V))

(59)

which is impossible Therefore


120572(V 119879 (V)) = 0 (60)

Suppose now that exist1205720isinA

existVisinFix(119879[119896]) 1198691205720(119879(V) V) gt 0 Then

by Definition 27(A) and property (60) using the fact that V =

119879[119896]

(V) = 119879[2119896]

(V) we get for 120578 isin 1 2 that

0 lt 1198691205720

(119879 (V) V) = 1198691205720

(119879[119896+1]

(V) 119879[2119896](V))

le

119896minus2sum

119898=1119862119898

12057201198691205720

(119879[119896+119898]

(V) 119879[119896+119898+1](V))

+119862119896minus21205720

1198691205720

(119879[2119896minus1]

(V) 119879[2119896](V))

le

119896minus2sum

119898=1119862119898

1205720sdot 119863

119871minusJ119862A1205781198831205720

(119879[119896+119898]

(V) 119879[119896+119898+1](V))

+119862119896minus21205720

sdot 119863119871minusJ119862A1205781198831205720

(119879[2119896minus1]

(V) 119879[2119896](V))

le

119896minus2sum

119898=1119862119898

1205720(

1205821205720

1198621205720

)

119896+119898

1198691205720

(V 119879 (V))

+119862119896minus21205720

(1205821205720

1198621205720

)

2119896minus1

1198691205720

(V 119879 (V)) = 0

(61)



120572(119879 (V) V) = 0 (62)

We see that (56) is a consequence of (60) and (62)

Part 2 Properties (57) and (58) hold We first observe that

forallVisinFix(119879[119896]) 119879 (V) = V (63)

in other words Fix(119879[119896]

) = Fix(119879) In fact if V isin Fix(119879[119896]

)

and 119879(V) = V then since the family P119862A = 119901

120572 120572 isin A is

separating on119883 we get that 119879(V) = V rArr exist120572isinA 119901

120572(119879(V) V) gt

0or119901120572(V 119879(V)) gt 0 In view ofTheorem 15 this implies119879(V) =

V rArr exist120572isinA 119869

120572(119879(V) V) gt 0 or 119869

120572(V 119879(V)) gt 0 However by

property (56) this is impossibleNext we see that forallVisinFix(119879[119896])=Fix(119879) 119869

120572(V V) = 0 In fact

by Definition 11(A) and property (56) we conclude thatforall120572isinA forallVisinFix(119879[119896]) 119869

120572(V V) le 119862

120572[119869

120572(V 119879(V))+119869

120572(119879(V) V)] = 0

Finally suppose that 119906 119908 isin Fix(119879) and 119906 = 119908 Thensince the family P

119862A = 119901120572 120572 isin A is separating on 119883

we get exist1205720isinA

1199011205720(119906 119908) gt 0 or 119901

1205720(119908 119906) gt 0 By applying

Theorem 15 this implies exist1205720isinA

1198691205720(119906 119908) gt 0 or 119869

1205720(119908 119906) gt

0 Consequently for 120578 isin 1 2 by Definition 27(A) weconclude that

exist1205720isinA

[1198691205720

(119906 119908) gt 0 1198691205720

(119906 119908)

= 1198691205720

(119879 (119906) 119879 (119908)) le119863119871minusJ119862A1205781198831205720

(119879 (119906) 119879 (119908))

le(1205821205720

1198621205720

)1198691205720

(119906 119908) lt 1198691205720

(119906 119908)] or [1198691205720

(119908 119906)

gt 0 1198691205720

(119908 119906) = 1198691205720

(119879 (119908) 119879 (119906))

le119863119871minusJ119862A1205781198831205720

(119879 (119908) 119879 (119906)) le(1205821205720

1198621205720

)1198691205720

(119908 119906)

lt 1198691205720

(119908 119906)]

(64)

which is impossible This gives that Fix(119879) is a singletonThus (57) and (58) hold

10 Examples of Spaces (119883P119862A)

Example 1 Let 119883 = [0 6] 120574 ge 81 and let 119901 1198832

rarr [0infin)

be of the form

119901 (119906 119908)

=

0 if 119906 ge 119908 119906 119908 cap (0 6) = 119906 119908

(119908 minus 119906)4 if 119906 lt 119908 119906 119908 cap (0 6) = 119906 119908

120574 if 119906 119908 cap (0 6) = 119906 119908

(65)

(1) (119883P81) P81 = 119901 is the quasi-triangular

space In fact

forall119906V119908isin119883 119901 (119906 119908) le 8 [119901 (119906 V) + 119901 (V 119908)] (66)

Inequality (66) is a consequence of Cases 1ndash6

Case 1 If 119906 V 119908 isin (0 6) and V le 119906 lt 119908 then 119901(119906 V) = 0 and119908 minus 119906 le 119908 minus V This gives 119901(119906 119908) = (119908 minus 119906)

4le (119908 minus V)4 lt

8(119908 minus V)4 = 8[119901(119906 V) + 119901(V 119908)]

Case 2 If 119906 V 119908 isin (0 6) 119906 lt 119908 and 119906 le V le 119908 then119901(119906 119908) =

(119908 minus 119906)4 and 119891(V0) = min

119906leVle119908119891(V) = (119908 minus 119906)4 where for

119906 le V le 119908 119891(V) = 8[119901(119906 V)+119901(V 119908)] = 8[(Vminus119906)4+(119908minus V)4]

and V0 = (119906 + 119908)2

Case 3 sup119906119908isin(06)119906lt119908119901(119906 119908) = sup

119906119908isin(06)119906lt119908(119908 minus 119906)4

=

64 = 1296 and sup119906119908isin(06)119906lt119908min

119906leVle1199088[119901(119906 V) + 119901(V 119908)] =

sup119906119908isin(06)119906lt119908min

119906leVle1199088[(V minus 119906)4+ (119908 minus V)4] = 8[(3 minus 0)4 +

(6 minus 3)4] = 8[81 + 81] = 1296


Case 4 If 119906 V 119908 isin (0 6) and 119906 lt 119908 le V then 119901(V 119908) = 0 and119908 minus 119906 le V minus 119906 This gives 119901(119906 119908) = (119908 minus 119906)

4le (V minus 119906)

4lt

8(V minus 119906)4= 8[119901(119906 V) + 119901(V 119908)]

Case 5 If 119906 119908 isin (0 6) 119906 lt 119908 and V isin 0 6 then 119901(119906 119908) le

1296 le 8[119901(119906 V) + 119901(V 119908)] = 8[120574 + 120574]

Case 6 If 119906 119908 cap (0 6) = 119906 119908 then forallVisin119883 119901(119906 119908) = 120574 lt

8120574 le 8[119901(119906 V) + 119901(V 119908)]

(2) P81 = 119901 is asymmetric Indeed we have that 0=

119901(5 1) = 119901(1 5) = 256 Therefore condition forall119906119908isin119883

119901(119906119908) = 119901(119908 119906) does not hold

(3)P81 = 119901 does not vanish on the diagonal Indeed

if 119906 isin 0 6 then 119901(119906 119906) = 120574 = 0 Therefore the conditionforall119906isin119883

119901(119906 119906) = 0 does not hold(4) For the constant sequence of the form (119906

119898= 3

119898 isin N) sub 119883 the sets LIM119871minusP81(119906119898 119898isinN)

and LIM119877minusP81(119906119898 119898isinN)

are notsingletons Indeed by (65) Remark 12 and Definition 16(B)we have that LIM119871minusP81

(119906119898 119898isinN)= [3 6] and LIM119877minusP81

(119906119898 119898isinN)= [0 3]

Example 2 Let119883 be a set (nonempty)119860 sub 119883119860 = 119860 = 119883120574 gt 0 and let 119901 119883

2rarr [0infin) be of the form

119901 (119906 119908) =

0 if 119860 cap 119906 119908 = 119906 119908

120574 if 119860 cap 119906 119908 = 119906 119908

(67)

(1) A pair (119883P11) P11 = 119901 is the quasi-trian-

gular space Indeed formula (67) yields forall119906V119908isin119883 119902(119906 119908) le

119902(119906 V) + 119902(V 119908) Otherwise exist1199060 V0 1199080isin119883

119902(1199060 1199080) gt 119902(1199060V0) + 119902(V0 1199080) It is clear that then 119902(1199060 1199080) = 120574 119902(1199060 V0) =

0 and 119902(V0 1199080) = 0 From this we see that 119860 cap 1199060 1199080 =

1199060 1199080 119860 cap 1199060 V0 = 1199060 V0 and 119860 cap V0 1199080 = V0 1199080This is impossible



119901(119906 119906) = 0 does not hold(3)P

11 = 119901 is symmetricThis follows from (67)(4) We observe that LIM119871minusP11

(119906119898 119898isinN)= LIM119877minusP11

(119906119898 119898isinN)= 119860 for

each sequence (119906119898

119898 isin N) sub 119860 We conclude this from (67)

Example 3 Let 119883 = [0 6] and let 119901 1198832

rarr [0infin) be of theform

119901 (119906 119908) =

0 if 119906 ge 119908

(119908 minus 119906)3 if 119906 lt 119908

(68)


space In fact forall119906V119908isin119883 119902(119906 119908) le 4[119902(119906 V) + 119902(V 119908)] holds

This is a consequence of Cases 1ndash3

Case 1 If V le 119906 lt 119908 then 119901(119906 V) = 0 119908 minus 119906 le 119908 minus V andconsequently 119901(119906 119908) = (119908 minus 119906)

3le (119908 minus V)3 lt 4(119908 minus V)3 =

4119901(V 119908) = 4[119901(119906 V) + 119901(V 119908)]

Case 2 If 119906 lt 119908 and 119906 le V le 119908 then 119902(119906 119908) = (119908 minus 119906)3 and

119891(V0) = min119906⩽V⩽119908119891(V) = (119908 minus 119906)

3 where V0 = (119906 + 119908)2 is

a minimum point of the map 119891(V) = 4[119901(119906 V) + 119901(V 119908)] =

4(119908 minus 119906)[1199082+ 119908119906 + 119906

2+ 3V2 minus 3V(119908 + 119906)]

Case 3 If 119906 lt 119908 le V then 119901(V 119908) = 0 and consequently119901(119906 119908) = (119908 minus 119906)


3lt 4(V minus 119906)

3= 4119901(119906 V) = 4[119901(119906

V) + 119901(V 119908)]



119901(119906119908) = 119901(119908 119906) does not hold

(3)P41 = 119901 vanishes on the diagonal In fact by (68)

it is clear that forall119906isin119883

119901(119906 119906) = 0(4) We observe that LIM119871minusP41

(119906119898 119898isinN)= [2 6] and

LIM119877minusP41(119906119898 119898isinN)

= [1 2] for sequence (119906119898

= 2 119898 isin N) We con-clude this from (68)

Example 4 Let 119883 = [0 6] and let P21 = 119901 where 119901

1198832

rarr [0infin) is of the form

119901 (119906 119908) =

0 if 119906 ge 119908

(119906 minus 119908)2 if 119906 lt 119908

(69)

Let

119864 = [0 3) cup (3 6] (70)

and let 120583 ge 364 andJ21 = 119869 where 119869 119883

2rarr [0infin) is

of the form

119869 (119906 119908) =

119901 (119906 119908) if 119864 cap 119906 119908 = 119906 119908

120583 if 119864 cap 119906 119908 = 119906 119908

(71)

(1)J21 is not symmetric In fact by (69)ndash(71) 119869(0 6) =

36 and 119869(6 0) = 0(2)J

21 = 119869 isin J119871(119883P21)

capJ119877(119883P21)

SeeTheorem 14

Remark 33 By Examples 1ndash4 it follows that the distances119901 defined by (65) and (67)ndash(69) and 119869 defined by (70) and(71) are not metrics ultra metrics quasi-metrics ultra quasi-metrics 119887-metrics partial metrics partial 119887-metrics pseu-dometrics (gauges) quasi-pseudometrics (quasi-gauges) andultra quasi-pseudometrics (ultra quasi-gauges)

11 Examples Illustrating Theorem 26

Example 1 Let119883 = [0 6] let 120574 gt 2048 be arbitrary and fixedand for 119906 119908 isin 119883 let

119901 (119906 119908)

=

0 if 119906 ge 119908 119906 119908 cap (0 6) = 119906 119908

(119908 minus 119906)4 if 119906 lt 119908 119906 119908 cap (0 6) = 119906 119908

120574 if 119906 119908 cap (0 6) = 119906 119908

(72)

Define the set-valued dynamic system (119883 119879) by

119879 (119906) =

[1 2] if 119906 isin [0 3) cup (4 6]

(4 6) if 119906 isin [3 4] (73)


Let119864 = [0 3) cup (4 6] (74)

and let 119869 119883 times 119883 rarr [0infin) be of the form

119869 (119906 119908) =

119901 (119906 119908) if 119864 cap 119906 119908 = 119906 119908

120574 if 119864 cap 119906 119908 = 119906 119908

(75)

(1) (119883P81) where P

81 = 119901 is the quasi-triangular space and J

81 = 119869 is the left and right familygenerated by P

81 This is a consequence of Definitions 7and 11 Example 1 andTheorem 14 we see that 120574 = 120583 gt 81

(2) (119883 119879) is a (119863 = D119871minusJ8112119883 = D

119877minusJ8112119883 120582 isin [2048

120574 1))-quasi-contraction on 119883 that is forall120582isin[20481205741) forall

119909119910isin1198838 sdot

119863(119879(119909) 119879(119910)) le 120582119869(119909 119910) where

119863 (119880119882) = maxsup119906isin119880

119869 (119906119882) sup119908isin119882

119869 (119880 119908)

119880119882 isin 2119883

(76)

Indeed we see that this follows from (73)ndash(76) and fromCases 1ndash4 below

Case 1 If119909 119910 isin [0 3)cup(4 6] then119879(119909) = 119879(119910) = [1 2] = 119880 sub

119864 and sup119906isin119880

inf119908isin119880

119869(119906 119908) = sup119906isin119880

119869(119906 119906) = 119901(119906 119906) =

0 = 0 Thus 4119863(119879(119909) 119879(119910)) = 0 le 120582119869(119909 119910)

Case 2 If 119909 isin [0 3)cup(4 6] and 119910 isin [3 4] then119879(119909) = [1 2] =

119880 sub 119864 119879(119910) = (4 6) = 119882 sub 119864 and sup119906isin119880

inf119908isin119882

119869(119906 119908) =

sup119906isin119880

inf119908isin119882

(119908 minus 119906)4 = sup

119906isin119880(4 minus 119906)

4= 81 and

sup119908isin119882

inf119906isin119880

119869(119906 119908) = sup119908isin119882

inf119906isin119880

(119908 minus 119906)4 =

sup119908isin119882

(119908 minus 2)4 = 256 Thus 8119863(119879(119909) 119879(119910)) = 2048 Onthe other hand 119910 notin 119864 which gives 119869(119909 119910) = 120574 Therefore8119863(119879(119909) 119879(119910)) le 120582119869(119909 119910) whenever 2048 le 120582120574 This gives2048120574 le 120582 lt 1 whenever 120574 gt max2048 81

Case 3 If 119909 isin [3 4] and 119910 isin [0 3) cup (4 6] then 119879(119909) =

(4 6) = 119880 sub 119864 and 119879(119910) = [1 2] = 119882 sub 119864 Hence weobtain sup

119906isinUinf119908isin119882

119869(119906 119908) = sup119906isin119880

inf119908isin119882

119901(119906 119908) =

sup119908isin119882

inf119906isin119880

119869(119906 119908) = sup119908isin119882

inf119906isin119880

119901(119906 119908) = 0Therefore 8119863(119879(119909) 119879(119910)) = 0 le 120582119869(119909 119910)

Case 4 If 119909 119910 isin [3 4] then 119879(119909) = 119879(119910) = (4 6) = 119880 sub 119864Therefore 41198631(119879(119909) 119879(119910)) = 0 le 120582119869(119909 119910)

(3) Property (26) holds that isforall119909isin119883

forall120573isin(0infin)

exist119910isin119879(119909)

119869(119909

119910) lt 119869(119909 119879(119909)) + 120573 Indeed this follows from Cases 1ndash4below

Case 1 If 1199090 = 0 and 1199100 = 1 isin 119879(1199090) = [1 2] then 119869(1199090 1199100) =

120574 119869(1199090 119879(1199090)) = inf119908isin[12]119869(1199090 119908) = 120574 and forall

120573isin(0infin)119869(1199090

1199100) lt 119869(1199090 119879(1199090)) + 120573

Case 2 If 1199090 isin (0 1] and 1199100 = 1 isin 119879(1199090) = [1 2] then119869(1199090 1199100) = 1 minus 1199090 119869(1199090 119879(1199090)) = inf

119908isin[12]119869(1199090 119908) = 1 minus 1199090and forall

120573isin(0infin)119869(1199090 1199100) lt 119869(1199090 119879(1199090)) + 120573

Case 3 If 1199090 isin (1 3) cup (4 6) and 1199100 = 1 isin 119879(1199090) = [1 2]then 119869(1199090 1199100) = 0 119869(1199090 119879(1199090)) = 0 and forall

120573isin(0infin)119869(1199090 1199100) lt

119869(1199090 119879(1199090)) + 120573

Case 4 If 1199090 isin [3 4] and 1199100 isin 119879(1199090) = (4 6) then 119869(1199090 1199100) =

120574 119901(1199090 119879(1199090)) = 120574 and forall120573isin(0infin)

119869(1199090 1199100) lt 119869(1199090 119879(1199090)) +

120573

Case 5 If 1199090 = 6 and 1199100 isin 119879(1199090) = [1 2] then 119869(1199090 1199100) = 120574119901(1199090 119879(1199090)) = 120574 and forall

120573isin(0infin)119869(1199090 1199100) lt 119869(1199090 119879(1199090)) + 120573

(4) (119883 119879) is left and right J81-admissible in each point

1199080isin 119883 In fact if1199080

isin 119883 and (119908119898

119898 isin 0cupN) are such thatforall119898isin0cupN 119908

119898+1isin 119879(119908

119898

) and lim119898rarrinfin

sup119899gt119898

119869(119908119898

119908119899

) =

0 (lim119898rarrinfin

sup119899gt119898

119869(119908119899

119908119898

) = 0) then forall119898ge2 119908

119898

isin [1 2]and consequently by (72) forall

119908isin[26)sub119883 lim119898rarrinfin

119901(119908119908119898

) =

0 (forall119908isin(01]sub119883 lim

119898rarrinfin119901(119908

119898

119908) = 0) Hence by (75)and (76) we get forall

119908isin[23)cup(46)sub119883 lim119898rarrinfin

119869(119908 119908119898

) =


119898rarrinfin119869(119908

119898

119908) = 0)(5) (119883 119879) is a left and right P

81-closed on 119883 Indeedlet (119909

119898 119898 isin N) sub 119879(119883) be a left (right)P

81-convergingsequence in 119883 (thus LIM119871minusP81

(119909119898 119898isinN)= (LIM119877minusP81

(119909119898 119898isinN)= ))

and having subsequences (V119898

119898 isin N) and (119906119898

119898 isin N)

satisfying forall119898isinN V

119898isin 119879(119906

119898) Then forall

119898ge2 119909119898

isin [1 2] 2 isin

119879(2) and 2 isin LIM119871minusP81(119909119898 119898isinN)

(1 isin 119879(1) and 1 isin LIM119877minusP81(119909119898 119898isinN)

)(6) All assumptions of Theorem 26 are satisfied This

follows from (1)ndash(5) in Example 1We conclude that Fix(119879) = [1 2] and we have shown the

following

Claim A 2 isin 119879(2) and 2 isin LIM119871minusP81(119908119898119898isin0cupN) for each 119908

0isin 119883

and for each dynamic process (119908119898

119898 isin 0 cup N) of thesystem (119883 119879)

Claim B 1 isin 119879(1) and 1 isin LIM119877minusP81(119908119898119898isin0cupN) for each 119908

0isin 119883



Example 2 Let 119883 P81 = 119901 and (119883 119879) be such as in

Example 1(1) For each 120582 isin [0 1) condition forall

119909119910isin1198838119863(119879(119909)

119879(119910)) le 120582119901(119909 119910) where 119863(119880119882) = maxsup119906isin119880

119901(119906119882) sup

119908isin119882119901(119880119908) 119880119882 isin 2119883 does not hold Suppose that


119909119910isin1198838119863(119879(119909) 119879(119910)) le 1205820119901(119909 119910) Letting 1199090 = 2

and 1199100 = 3 it can be shown that 119901(1199090 1199100) = 1 119879(1199090) =

[1 2] = 119880 119879(1199100) = (4 6) = 119882 sup119906isin[12]119901(119906 (4 6)) =

sup119906isin[12](4 minus 119906)

4= 34 = 81 and sup

119908isin(46)119901([1 2] 119908) =

sup119908isin(46)(119908 minus 2)4 = 44 = 256 Therefore 2048 = 8119863(119879(1199090)

119879(1199100)) = 8max81 256 le 1205820119901(1199090 1199100) = 1205820 which isabsurd

Remark 34 Wemake the following remarks about Examples1 and 2 and Theorem 26 (a) By Example 1 we observe thatwe may apply Theorem 26 for set-valued dynamic systems(119883 119879) in the left and right quasi-triangular space (119883P

119862A)

with left and right family J119862A generated by P

119862A whereJ

119862A = P119862A (b) By Example 2 we note however that

we do not apply Theorem 26 in the quasi-triangular space(119883P

119862A)whenJ119862A = P

119862A (c) From (a) and (b) it followsthat inTheorem 26 the existence of left (right) familiesJ

119862A

generated byP119862A and such thatJ

119862A = P119862A are essential


Example 3 Let 119883 = (0 6) 120574 gt 0 and

119860 = 1198601 cup1198602 1198601 = (0 2] 1198602 = [4 6) (77)

Let 119901 1198832

rarr [0infin) be of the form

119901 (119906 119908) =

0 if 119860 cap 119906 119908 = 119906 119908

120574 if 119860 cap 119906 119908 = 119906 119908

(78)

and letJ11 = P

11 = 119901 Define the set-valued dynam-ic system (119883 119879) by

119879 (119906) =

1198602 for 119906 isin (0 3)

119860 for 119906 = 3

1198601 for 119906 isin (3 6)

(79)

(1) (119883P11) is quasi-triangular space See Example 2

Section 11(2) (119883 119879) is a (D

119871minusP1112119883 120582 isin [0 1))-quasi-contraction

on 119883 that is forall120582isin[01) forall

119909119910isin119883119863

119871minusP1112119883 (119879(119909) 119879(119910)) le 120582119901(119909

119910) Indeed if 119909 119910 isin 119883 then by (77)ndash(79) 119879(119909) 119879(119910) sub 119860

and maxsup119906isin119879(119909)

119901(119906 119879(119910)) sup119908isin119879(119910)

119901(119879(119909) 119908) = 0(3) Property (16) holds that is forall

119909isin119883forall120573isin(0infin)

exist119910isin119879(119909)

119901(119909 119910) lt 119901(119909 119879(119909)) + 120573 Indeed this follows fromCases 1ndash3 below

Case 1 Let 1199090 isin (0 3) and 120573 isin (0infin) be arbitrary and fixedIf 1199100 isin 119879(1199090) = 1198602 then by (78)

119901 (1199090 1199100) = 119901 (1199090 119879 (1199090))

=

0 if 1199090 isin 1198601

120574 for 1199090 isin (0 3) 1198601

(80)

Therefore 119901(1199090 1199100) lt 119901(1199090 119879(1199090)) + 120573

Case 2 Let 1199090 = 3 and let 120573 isin (0infin) be arbitrary and fixed If1199100 isin 119879(1199090) = 119860 then by (78) 119901(1199090 1199100) = 119901(1199090 119879(1199090)) = 120574Therefore 119901(1199090 1199100) lt 119901(1199090 119879(1199090)) + 120573


119901 (1199090 1199100) = 119901 (1199090 119879 (1199090))

=

0 if 1199090 isin 1198602

120574 for 1199090 isin (3 6) 1198602

(81)

Therefore 119901(1199090 1199100) lt 119901(1199090 119879(1199090)) + 120573(4) (119883 119879) is left and rightP

11-admissible in119883Assum-ing that 119908

0isin 119883 is arbitrary and fixed we prove that if the

dynamic process (119908119898

119898 isin 0 cup N) of (119883 119879) startingat 119908

0 is such that lim119898rarrinfin

sup119899gt119898

119901(119908119898

119908119899

) = 0 thenexist119908isin119883

lim119898rarrinfin

119901(119908119908119898

) = 0 Indeed if 1199080isin 119883 then by

(79)forall119898⩾1 119908

119898

isin 119879(119908119898minus1

) sub 119860 and by (78) we immediatelyget 119860 = LIM119871minusP11

(119908119898119898isin0cupN) = LIM119877minusP11

(119908119898119898isin0cupN)

(5) Set-valued dynamic system (119883 119879[2]

) is a left and rightP

11-closed on 119883 Indeed if (119909119898

119898 isin N) sub 119879[2]

(119883) = 119860

is a left or rightP11-converging sequence in119883 and having

subsequences (V119898

119898 isin N) and (119906119898

119898 isin N) satisfyingforall119898isinN V

119898isin 119879(119906

119898) then by (77)ndash(79) we have that

exist1198980isinN

forall119898⩾1198980

119909119898

isin 119860 119860 = LIM119871minusP11(119909119898 119898isinN)

= LIM119877minusP11(119909119898 119898isin0cupN)

and Fix(119879[2]) = 119860

(6) For (119883P11)P11 = 119901 J

11 = P11 and

(119883 119879) defined by (77)ndash(79) all assumptions ofTheorem 26 aresatisfiedThis follows from (1)ndash(5) in Example 3

We conclude that Fix(119879[2]) = 119860 and we claim that if 1199080

isin

119883 1199081isin 119879(119908

0) and 119908

2= 119906 isin 119879(119908

1) are arbitrary and fixed

and forall119898⩾3 119908

119898

= 119906 then sequence (119908119898

119898 isin 0 cup N) is adynamic process of119879 starting at1199080 and left and rightP

11-converging to each point of 119860 We observe also that Fix(119879) =


1198832


119901 (119906 119908) =

0 if 119906 ge 119908

(119906 minus 119908)2 if 119906 lt 119908

(82)


119879 (119906) = ([0 3) cup (3 6]) 119906 for 119906 isin [0 6] (83)

Let

119864 = [0 3) cup (3 6] (84)


2rarr [0infin) is

of the form

119869 (119906 119908) =

119901 (119906 119908) if 119864 cap 119906 119908 = 119906 119908

120583 if 119864 cap 119906 119908 = 119906 119908

(85)

(1)J21 is not symmetric In fact by (82) (84) and (85)

119869(0 6) = 36 and 119869(6 0) = 0(2)J

21 = 119869 isin J119871(119883P21)

capJ119877(119883P21)

SeeTheorem 14

(3) (119883 119879) is a (119863 = D119871minusJ2112119883 120582 isin [0 1))-contraction on

119883 that is forall119909119910isin119883

2 sdot 119863(119879(119909) 119879(119910)) le 120582119869(119909 119910) where 120582 isin

[0 1) and

119863(119880119882) = maxsup119906isin119880

119869 (119906119882) sup119908isin119882

119869 (119880 119908)

119880119882 isin 2119883

(86)

Indeed we see that this follows from (1) (2) in Example 4and from Cases 1ndash4 below


Case 1 Let 119909 119910 isin [0 3)cup(3 6]Then 119909 119910 isin 119864119879(119909) = ([0 3)cup(3 6])119909 = 119880 sub 119864 and119879(119910) = ([0 3)cup(3 6])119910 = 119882 sub 119864If 119906 isin 119880 then we have 119882 = 119882

119906

cup 119882119906and

inf119908isin119882

119869 (119906 119908)

le

inf119908isin119882119906119902 (119906 119908) = 0 if 119882

119906

= 119908 isin 119882 119906 ge 119908 =

inf119908isin119882119906

(119906 minus 119908)2= 0 if 119882

119906= 119908 isin 119882 119906 lt 119908 =

(87)

and if 119908 isin 119882 then we have 119880 = 119880119908

cup 119880119908and

inf119906isin119880

119869 (119906 119908)

le

inf119906isin119880119908119902 (119906 119908) = 0 if 119880

119906

= 119906 isin 119880 119906 ge 119908 =

inf119906isin119880119908

(119906 minus 119908)2= 0 if 119880

119906= 119906 isin 119880 119906 lt 119908 =

(88)

By (86) 2119863(119879(119909) 119879(119910)) = 0 le 120582119869(119909 119910)

Case 2 If 119909 = 119910 = 3 then 119869(119909 119910) = 120583 and 119879(119909) =

119879(119910) = [0 3) cup (3 6] = 119880 sub 119864 Therefore 2119863(119879(119909) 119879(119910)) =

2119863(119880119880) = 0 le 120582119869(119909 119910)

Case 3 If 119909 isin [0 3) cup (3 6] and 119910 = 3 then 119909 isin 119864 119910 notin 119864119869(119909 119910) = 120583 119879(119909) = ([0 3) cup (3 6]) 119909 = 119880 sub 119864 and 119879(119910) =

[0 3) cup (3 6] = 119882 sub 119864 We see that sup119906isin119880

inf119908isin119882

119869(119906 119908) =

0 since if 119906 isin 119880 then also 119908 = 119906 isin 119882 and inf119908isin119882

119869(119906 119908) =

119902(119906 119906) = 0 Next we see that sup119908isin119882

inf119906isin119880

119869(119906 119908) = 0since if 119908 isin 119882 then 119880 = 119880

119908

cup 119880119908and

inf119906isin119880

119869 (119906 119908)

le

inf119906isin119880119908119902 (119906 119908) = 0 if 119880

119906

= 119906 isin 119880 119906 ge 119908 =

inf119906isin119880119908

(119906 minus 119908)2= 0 if 119880

119906= 119906 isin 119880 119906 lt 119908 =

(89)

Thus 2119863(119879(119909) 119879(119910)) = 0 ⩽ 120582119869(119909 119910)

Case 4 If 119909 = 3 and 119910 isin [0 3) cup (3 6] then 119909 notin 119864 119910 isin 119864119869(119909 119910) = 120583 119879(119909) = [0 3) cup (3 6] = 119880 sub 119864 119879(119910) = ([0 3) cup

(3 6]) 119910 = 119882 sub 119864 and sup119906isin119880

inf119908isin119882

119869(119906 119908) = 0 sincefor 119906 isin 119880

inf119908isin119882

119869 (119906 119908)

le

inf119908isin119882119906119902 (119906 119908) = 0 if 119882

119906

= 119908 isin 119882 119906 ge 119908 =

inf119908isin119882119906

(119906 minus 119908)2= 0 if 119882

119906= 119908 isin 119882 119906 lt 119908 =

(90)

and sup119908isin119882

inf119906isin119880

119869(119906 119908) = 0 since inf119906isin119880

119869(119906 119908) = 119869(119908119908) = 0 for 119908 isin 119882 Thus 2119863(119879(119909) 119879(119910)) = 0 le 120582119869(119909 119910)

(4) Property (26) holds that is forall119909isin119883


exist119910isin119879(119909)

119869(119909


Case 1 Let 1199090 isin [0 3) and 120574 isin (0infin) be arbitrary and fixedIf 1199100 isin 119879(1199090) = ([0 3) cup (3 6]) 1199090 = 119882 is such that

1199090 lt 1199100 lt 3 then 119869(1199090 1199100) = (1199090 minus 1199100)2 and 119869(1199090 119879(1199090)) =

inf119908isin119882

119869(1199090 119908) = 0 since

inf119908isin119882

119869 (1199090 119908)

le

inf119908isin1198821199090119902 (1199090 119908) = 0 if 119882

1199090 = 119908 isin 119882 1199090 ge 119908 =

inf119908isin1198821199090

(1199090 minus 119908)2= 0 if 119882

1199090= 119908 isin 119882 1199090 lt 119908 =

(91)

Then we see that 119869(1199090 1199100) = (1199090 minus 1199100)2lt 120574 implies 1199100 lt 1199090 +

12057412 From this we conclude that if1199100 isin (1199090min3 1199090+120574

12)

then 119869(1199090 1199100) lt 119869(1199090 119879(1199090)) + 120574

Case 2 Let 1199090 = 3 Assume that 1199100 isin 119879(1199090) = [0 3) cup (3 6]is arbitrary and fixed Then 119869(1199090 1199100) = 120583 119869(1199090 119879(1199090)) =

inf119908isin[03)cup(36]119869(1199090 119908) = 120583 and for each 120574 isin (0infin)

119869(1199090 1199100) lt 119869(1199090 119879(1199090)) + 120574

Case 3 Let 1199090 isin (3 6] and 120574 isin (0infin) be arbitrary and fixedIf 1199100 isin 119879(1199090) = ([0 3) cup (3 6]) 1199090 = 119882 is such that 3 lt

1199100 lt 1199090 then 119869(1199090 1199100) = 0 and analogously as in Case 1 weget 119869(1199090 119879(1199090)) = inf

119908isin119882119869(1199090 119908) = 0 Therefore 119869(1199090 1199100) lt

119869(1199090 119879(1199090)) + 120574

(5) (119883 119879) is left J21-admissible in 119883 Assuming that

1199080

isin 119883 is arbitrary and fixed we prove that if thedynamic process (119908

119898



sup119899gt119898

119869(119908119898

119908119899


lim119898rarrinfin

119869(119908 119908119898

) = 0 We consider the followingcases

Case 1 If 1199080isin [0 3) cup (3 6] then 119908

1isin 119879(119908

0) = ([0 3) cup

(3 6]) 1199080 and forall

119898ge2 119908119898

isin 119879(119908119898minus1

) sub [0 3) cup (3 6] andusing (82) we immediately get 6 isin LIM119871minusJ21

(119908119898119898isin0cupN)

Case 2 If 1199080= 3 then 119908

1isin 119879(119908

0) = [0 3) cup (3 6] 1199082

isin

119879(1199081) = ([0 3) cup (3 6]) 119908

1 and forall

119898⩾3 119908119898

isin 119879(119908119898minus1

) sub

[0 3) cup (3 6] and using (82) we also immediately get 6 isin

LIM119871minusJ21(119908119898119898isin0cupN)

This shows that 6 isin LIM119871minusJ21(119908119898119898isin0cupN) for each 119908

0isin 119883 and

for each dynamic process (119908119898

119898 isin 0 cup N) of the system(119883 119879) we see that here property lim

119898rarrinfinsup

119899gt119898119869(119908

119898

119908119899

) =

0 of (119908119898

119898 isin 0 cup N) is not required(6) Set-valued dynamic system (119883 119879

[2]) is a left P

21-quasi-closed on 119883 Indeed if (119909

119898 119898 isin N) sub 119879

[2](119883) =

[0 3) cup (3 6] is a left P21-converging sequence in 119883 and

having subsequences (V119898

119898 isin N) and (119906119898

119898 isin N)


119898isin 119879(119906

119898) then by (83) we have that

exist1198980isinN

forall119898ge1198980

119909119898

isin [0 3) cup (3 6] Therefore in particular6 isin LIM119871minusP21

(119909119898 119898isinN)and 6 isin 119879

[2](6)

(7) ForP21 = 119901J

21 = 119869 and (119883 119879) defined by(82)ndash(85) all assumptions of Theorem 26 in the case of ldquoleftrdquoare satisfied This follows from (1)ndash(6) in Example 4

We conclude that Fix(119879[2]) = [0 3) cup (3 6] and we claim

that 6 isin 119879[2]

(6) and that 6 isin LIM119871minusP21(119908119898119898isin0cupN) for each 119908

0isin 119883



119898 isin 0 cup N) of thesystem (119883 119879) We observe also that Fix(119879) =

12 Example Illustrating Theorem 32

Example 1 Let 119883 = (0 6) 119860 and J11 = P

11 = 119901

be as in Example 3 Define the single-valued dynamic system(119883 119879) by

119879 (119906) =

4 for 119906 isin (0 3)

2 for 119906 isin [3 6) (92)

(1) (119883 119879) is a (D119871minusP111119883 120582 isin [0 1))-quasi-contraction


119909119910isin119883119863

119871minusP111119883 (119879(119909) 119879(119910)) =

max119901(119879(119909) 119879(119910)) 119901(119879(119910) 119879(119909)) le 120582119901(119909 119910) and Fix(119879) =

Indeed we see that if 119909 119910 isin 119883 then 119879(119909) 119879(119910) isin 119860 andby (77) and (78) 119863119871minusP11

1119883 (119879(119909) 119879(119910)) = 0 le 120582119901(119909 119910)(2) (119883 119879) is left and rightP

11-admissible in119883 Assumethat1199080

isin 119883 is arbitrary and fixed (119908119898

119898 isin 0cupN) satisfiesforall119898isin0cupN 119908

119898+1= 119879(119908

119898


sup119899gt119898

119901(119908119898

119908119899

) =

0Then by (92) and (78) we have forall119898isinN 119908

119898

isin 119860 This gives119860 = LIM119871minusP11


(119908119898119898isin0cupN)

(3) Single-valued dynamic system (119883 119879[2]

) is a left andright P


119898 isin N) sub

119879[2]

(119883) = 2 4 is a left P11-converging sequence in 119883


119898 isin N) and (119906119898

119898 isin N)


119898= 119879

[2](119906

119898) then by (77) (78) and (92)

we have that 119860 = LIM119871minusP11(119909119898 119898isinN)

In particular 2 = 119879[2]

(2) isin

LIM119871minusP11(119909119898 119898isinN)

and 4 = 119879[2]

(4) isin LIM119871minusP11(119909119898 119898isinN)

(4) Property (56) holds Indeed we have forallVisinFix(119879[2])=24

119901(V 119879(V)) = 119901(119879(V) V) = 0 since 119879(2) = 4 119879(4) = 2 and119879(2 4) = 2 4 sub 119860

(5) P11 = 119901 is not separating on 119883 Indeed if 119906 119908 isin

119883119860 then 119901(119906 119908) = 119901(119908 119906) = 120574 gt 0(6) For P

11 = 119901 (119883 119879) and J11 = P

11defined by (77) (78) and (79) parts (B1) and (B2) ofTheorem 32 hold but part (B3) of Theorem 32 does not holdThis follows from (1)ndash(5) in Example 1

13 Concluding Remarks

Remark 1 In Theorems 5 and 6 the following play animportant role (i) Distances 119889 and 119867

119889 as metrics satisfyconditions (A) of Definition 1 on119883 andCB(119883) respectively(ii) (119883 119889) and (CB(119883)119867

119889

) asmetric spaces are topologicaland Hausdorff spaces and the completeness of (119883 119889) impliescompleteness of (CB(119883)119867

119889

) (iii) The continuity of 119889 and119867

119889 on 119883 times 119883 and CB(119883) times CB(119883) respectively (iv)The continuity of maps 119879 (119883 119889) rarr (119883 119889) and 119879

(119883 119889) rarr (CB(119883)119867119889

) (as consequences of contractiveproperties defined in (1) and (3) resp) (v) InTheorem 6 theassumption that for each 119909 isin 119883 119879(119909) isin CB(119883)

Remark 2 Conclusions in Theorems 5 and 6 concern onlyfixed points but not periodic points this is a consequence

of separability of spaces (119883 119889) and (CB(119883)119867119889

) and alsocontinuity of 119879

Remark 3 In Theorems 26 and 32 properties concening thespaces andmaps such as mentioned above generally need nothold since spaces (119883P

119862A) with left (right) families J119862A

generated by P119862A are very general which is an obstruction

to use Nadlerrsquos and Banachrsquos reasoning Theorems 26 and 32show how to rectify this situation and are obtained withoutrestrictively required assumptions andwith conclusionsmoreprofound as in the well known results of this sort existing inthe literature

Conflict of Interests

The author declares that he has no conflict of interestsregarding the publication of this paper

References

[1] J-P Aubin and J Siegel ldquoFixed points and stationary pointsof dissipative multivalued mapsrdquo Proceedings of the AmericanMathematical Society vol 78 no 3 pp 391ndash398 1980

[2] J P Aubin and I Ekeland Applied Nonlinear Analysis JohnWiley amp Sons New York NY USA 1984

[3] J-P Aubin and H Frankowska Set-Valued Analysis vol 2 ofSystems amp Control Foundations amp Applications BirkhauserBoston Mass USA 1990

[4] G X-Z Yuan KKM Theory and Applications in NonlinearAnalysis Marcel Dekker New York NY USA 1999

[5] A C van RoovijNon Archimedean Functional Analysis MarcelDekker New York NY USA 1978

[6] I A Bakhtin ldquoThe contraction mapping principle in almostmetric spacerdquo Functional Analysis Ulrsquoyanovsk State PedagogicalUniversity Ulrsquoyanovsk Russia vol 30 pp 26ndash37 1989

[7] S Czerwik ldquoNonlinear set-valued contraction mappings inb-metric spacesrdquo Atti del Seminario Matematico e FisicodellrsquoUniversita di Modena vol 46 pp 263ndash276 1998

[8] S G Matthews ldquoPartial metric topologyrdquo Annals of the NewYork Academy of Sciences vol 726 pp 183ndash197 1994 Proceed-ings of the 8th Summer Conference on General Topology andApplication

[9] S Shukla ldquoPartial b-metric spaces and fixed point theoremsrdquoMediterranean Journal of Mathematics vol 11 no 2 pp 703ndash711 2014

[10] W A Wilson ldquoOn quasi-metric spacesrdquo American Journal ofMathematics vol 53 no 3 pp 675ndash684 1931

[11] H-P A Kunzi and O O Otafudu ldquoThe ultra-quasi-metricallyinjective hull of a T

0-ultra-quasi-metric spacerdquo Applied Cate-

gorical Structures vol 21 pp 651ndash670 2013[12] J DugundjiTopology AllynampBacon BostonMass USA 1966[13] I L Reilly ldquoQuasi-gauge spacesrdquo Journal of the London Mathe-

matical Society Second Series vol 6 pp 481ndash487 1973[14] S Banach ldquoSur les operations dans les ensembles abstraits

et leurs applications aux equations integralesrdquo FundamentaMathematicae vol 3 pp 133ndash181 1922

[15] S BNadler ldquoMulti-valued contractionmappingsrdquoNotices of theAmerican Mathematical Society vol 14 p 930 1967

[16] S B Nadler Jr ldquoMulti-valued contraction mappingsrdquo PacificJournal of Mathematics vol 30 pp 475ndash488 1969


[17] M M Deza and E Deza Encyclopedia of Distances SpringerBerlin Germany 2nd edition 2013

[18] W A Kirk and N Shahzad Fixed Point Theory in DistanceSpaces Springer Berlin Germany 2014

[19] J T Markin ldquoFixed point theorems for set valued contractionsrdquoNotices of the American Mathematical Society vol 15 p 9301968

[20] J T Markin ldquoA fixed point theorem for set valued mappingsrdquoBulletin of the American Mathematical Society vol 74 pp 639ndash640 1968

[21] V Berinde andM Pacurar ldquoThe role of the Pompeiu-Hausdorffmetric in fixed point theoryrdquo Creative Mathematics and Infor-matics vol 22 no 2 pp 143ndash150 2013

[22] D Tataru ldquoViscosity solutions of Hamilton-Jacobi equationswith unbounded nonlinear termsrdquo Journal of MathematicalAnalysis and Applications vol 163 no 2 pp 345ndash392 1992

[23] O Kada T Suzuki and W Takahashi ldquoNonconvex minimiza-tion theorems and fixed point theorems in complete metricspacesrdquoMathematica Japonica vol 44 no 2 pp 381ndash391 1996

[24] L-J Lin and W-S Du ldquoEkelandrsquos variational principle mini-max theorems and existence of nonconvex equilibria in com-plete metric spacesrdquo Journal of Mathematical Analysis andApplications vol 323 no 1 pp 360ndash370 2006

[25] T Suzuki ldquoGeneralized distance and existence theorems incomplete metric spacesrdquo Journal of Mathematical Analysis andApplications vol 253 no 2 pp 440ndash458 2001

[26] J S Ume ldquoExistence theorems for generalized distance oncomplete metric spacesrdquo Fixed Point Theory and Applicationsvol 2010 Article ID 397150 21 pages 2010

[27] I Valyi ldquoA general maximality principle and a fixed pointtheorem in uniform spacerdquo Periodica Mathematica Hungaricavol 16 no 2 pp 127ndash134 1985

[28] K Włodarczyk ldquoHausdorff quasi-distances periodic and fixedpoints for Nadler type set-valued contractions in quasi-gaugespacesrdquo Fixed Point Theory and Applications vol 2014 no 1article 239 2014

[29] K Wlodarczyk and R Plebaniak ldquoMaximality principle andgeneral results of Ekeland and Caristi types without lowersemicontinuity assumptions in cone uniform spaces with gen-eralized pseudodistancesrdquo Fixed Point Theory and Applicationsvol 2010 Article ID 175453 p 35 2010

[30] K Włodarczyk and R Plebaniak ldquoDynamic processes fixedpoints endpoints asymmetric structures and investigationsrelated to Caristi Nadler and Banach in uniform spacesrdquoAbstract and Applied Analysis vol 2015 Article ID 942814 16pages 2015

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Definition 1 Let 119883 be a (nonempty) set and let 119901 1198832

rarr

[0infin)(A) (119883 119901) is calledmetric if (i)forall

119906119908isin119883119901(119906 119908) = 0 iff 119906=

119908 (ii) forall119906119908isin119883

119901(119906 119908) = 119901(119908 119906) and (iii) forall119906V119908isin119883 119901(119906

119908) le 119901(119906 V) + 119901(V 119908)(B) (See [5]) (119883 119901) is called ultra metric if (i) forall

119906119908isin119883


119901(119906 119908) = 119901(119908 119906) and(iii) forall

119906V119908isin119883 119901(119906 119908) le max119901(119906 V) 119901(V 119908)(C) (See [6 7]) (119883 119901) is called 119887-metric with parameter

119862 isin [1infin) if (i) forall119906119908isin119883


119901(119906 119908) = 119901(119908 119906) and (iii) forall119906V119908isin119883 119901(119906 119908) le 119862[119901(119906 V) +

119901(V 119908)](D) (See [8]) (119883 119901) is called partial metric if (i) forall

119906119908isin119883

119906 = 119908 iff 119901(119906 119906) = 119901(119906 119908) = 119901(119908119908) (ii) forall119906119908isin119883

119901(119906

119906) le 119901(119906 119908) (iii) forall119906119908isin119883

119901(119906 119908) = 119901(119908 119906) and (iv)forall119906V119908isin119883 119901(119906 119908) le 119901(119906 V) + 119901(V 119908) minus 119901(V V)(E) (See [9]) (119883 119901) is called partial 119887-metric with param-

eter 119862 isin [1infin) if (i) forall119906119908isin119883

119906 = 119908 iff 119901(119906 119906) = 119901(119906

119908) = 119901(119908119908) (ii) forall119906119908isin119883

119901(119906 119906) le 119901(119906 119908) (iii) forall119906119908isin119883

119901(119906 119908) = 119901(119908 119906) and (iv) forall119906V119908isin119883 119901(119906 119908) le 119862[119901(119906 V) +

119901(V 119908)] minus 119901(V V)(F) (See [10]) (119883 119901) is called quasi-metric if (i) forall

119906119908isin119883

119901(119906 119908) = 0 iff 119906 = 119908 and (ii) forall119906V119908isin119883 119901(119906 119908) le 119901(119906 V) +

119901(V 119908)(G) (119883 119901) is called ultra quasi-metric if (i) forall

119906119908isin119883119901(119906

119908) = 0 iff 119906 = 119908 and (ii) forall119906V119908isin119883 119901(119906 119908) le max119901(119906 V)

119901(V 119908)(H) The distance 119901 is called pseudometric (or the gauge)

on119883 if (i) forall119906isin119883

119901(119906 119906) = 0 (ii) forall119906119908isin119883

119901(119906 119908) = 119901(119908 119906)and (iii) forall

119906V119908isin119883 119901(119906 119908) le 119901(119906 V) + 119901(V 119908)(I) The distance 119901 is called quasi-pseudometric (or the

quasi-gauge) on 119883 if (i) forall119906isin119883

119901(119906 119906) = 0 and (ii) forall119906V119908isin119883

119901(119906 119908) le 119901(119906 V) + 119901(V 119908)(J) (See [11]) The distance 119901 is called ultra quasi-

pseudometric (or the ultra quasi-gauge) on119883 if (i) forall119906isin119883

119901(119906119906) = 0 and (ii) forall

119906V119908isin119883 119901(119906 119908) le max119901(119906 V) 119901(V 119908)

Definition 2 (see [12]) Let 119883 be a (nonempty) set and letAbe an index set

(A) Each family D = 119889120572

120572 isin A of pseudometrics119889120572

1198832

rarr [0infin) 120572 isin A is called gauge on 119883 The gaugeD = 119889

120572 120572 isin A on 119883 is called separating if forall

119906119908isin119883119906 =

119908 rArr exist120572isinA 119889

120572(119906 119908) gt 0

(B) Let the familyD = 119889120572 120572 isin A be separating gauge

on 119883 The topology T(D) having as a subbase the familyB(D) = 119861(119906 119889

120572 120576

120572) 119906 isin 119883 120576


119861(119906 119889120572 120576

120572) = V isin 119883 119889

120572(119906 V) lt 120576

120572 with 119906 isin 119883 120576

120572gt 0 and

120572 isin A is called topology induced by D on 119883 the topologyT(D) is Hausdorff

(C) A topological space (119883T) such that there is aseparating gauge D on 119883 with T = T(D) is called a gaugespace and is denoted by (119883D)

Definition 3 (see [13]) Let 119883 be a (nonempty) set and let Abe an index set

(A) Each family P = 119901120572 120572 isin A of quasi-pseudom-

etrics 119901120572 119883

2rarr [0infin) 120572 isin A is called quasi-gauge on 119883

(B) Let the family P = 119901120572

120572 isin A be quasi-gaugeon 119883 The topology T(P) having as a subbase of the familyB(P) = 119861(119906 119901

120572 120576

120572) 119906 isin 119883 120576


119861(119906 119901120572 120576

120572) = V isin 119883 119901

120572(119906 V) lt 120576

120572 with 119906 isin 119883 120576

120572gt 0 and

120572 isin A is called topology induced byP on 119883(C) A topological space (119883T) such that there is a quasi-

gauge P on 119883 with T = T(P) is called quasi-gauge spaceand is denoted by (119883P)

Remark 4 (see [13 Theorems 42 and 26]) Each quasi-uniform space and each topological space is the quasi-gaugespace

There is a growing literature concerning set-valued andsingle-valued dynamic systems in the above defined distancespaces These studies contain also various extensions of theBanach [14] and Nadler [15 16] theorems Of course thereis a huge literature on this topic For some such spaces andtheorems in these spaces see for example M M Deza andE Deza [17] Kirk and Shahzad [18] and references therein

Recall that the first convergence existence approxima-tion uniqueness and fixed point result concerning single-valued contractions in complete metric spaces were obtainedby Banach in 1922 [14]

Theorem 5 (see [14]) Let (119883 119889) be a complete metric space If119879 119883 rarr 119883 and

exist0le120582lt1 forall119909119910isin119883

119889 (119879 (119909) 119879 (119910)) le 120582119889 (119909 119910) (1)

then the following are true (i)119879 has a unique fixed point119908 in119883

(ie there exists119908 isin 119883 such that119908 = 119879(119908) and Fix(119879) = 119908)and (ii) for each 119908

0isin 119883 the sequence (119879

[119898]

(1199080) 119898 isin N)

converges to 119908

The Pompeiu-Hausdorff metric 119867119889 on the class of all

nonempty closed and bounded subsetsCB(119883) of the metricspace (119883 119889) is defined as follows

119867119889

(119880119882) = maxsup119906isin119880

119889 (119906119882) sup119908isin119882

119889 (119908119880)

119880119882 isin CB (119883)

(2)

where for each 119909 isin 119883 and 119881 isin CB(119883) 119889(119909 119881) =

infVisin119881119889(119909 V) Using Pompeiu-Hausdorffmetric new contrac-tions were received by Nadler in 1967 and 1969 [15 16] as atool to study the existence of fixed points of set-valued mapsin complete metric spaces

Theorem 6 (see [15] [16 Theorem 5]) Let (119883 119889) be acomplete metric space If 119879 119883 rarr CB(119883) and


119909119910isin119883119867

119889

(119879 (119909) 119879 (119910)) le 120582119889 (119909 119910) (3)

then Fix(119879) = (ie there exists 119908 isin 119883 such that 119908 isin 119879(119908))

Markin [19 20] gave a slighty defferent version ofTheorem 6

Our primary interest is to construct new very general dis-tance spaces deliver new contractive set-valued and single-valued dynamic systems in these distance spaces present
















119862A = 119901120572

1198832

rarr



119906V119908isin119883 119901120572(119906 119908) le119862

120572[119901

120572(119906 V) + 119901

120572(V 119908)] (4)





forall119906119908isin119883

119906 =119908


120572(119906 119908) gt 0or119901

120572(119908 119906) gt 0

(5)


120572isinA

forall119906119908isin119883

119901minus1120572

(119906 119908) = 119901120572(119908 119906) then forall


minus1120572

(119906

119908) le 119862120572[119901

minus1120572

(119906 V) + 119901minus1120572


119862A) Pminus1119862A = 119901

minus1120572

1198832

rarr [0infin)



ces 119901120572

1198832



119901120572(119906 119906) = 0 or


119906119908isin119883119901

120572(119906 119908) = 119901

120572(119908 119906) or forall


120572(119906

119908) le 119901120572(119906 V) + 119901


Section 10



A


120572 119883

2rarr [0infin)



119906V119908isin119883 119897120572(119906 119908)

le119862120572max 119897

120572(119906 V) 119897

120572(V 119908)

(6)




120572 119883

2rarr [0infin)



119906V119908isin119883 119904120572(119906 119908) le119862

120572[119904

120572(119906 V) + 119904

120572(V 119908)]

minus 119904120572(V V)

(7)






119862A)






(A) The family J119862A = 119869


119869120572

1198832


119862A if


119906V119908isin119883 119869120572(119906 119908) le 119862

120572[119869

120572(119906 V) + 119869

120572(V 119908)]

and furthermore



119898 isin N) and (119908119898



119898rarrinfin

sup119899gt119898

119869120572(119906

119898 119906

119899) = 0 (8)


119898rarrinfin

sup119899gt119898

119869120572(119906

119899 119906

119898) = 0) (9)


119898rarrinfin

119869120572(119908

119898 119906

119898) = 0 (10)


119898rarrinfin

119869120572(119906

119898 119908

119898) = 0) (11)

the following holds


119898rarrinfin

119901120572(119908

119898 119906

119898) = 0 (12)


119898rarrinfin

119901120572(119906

119898 119908

119898) = 0) (13)

(B) J119871(119883P119862A)

(J119877(119883P119862A)




J119871(119883P119862A)

cap J119877(119883P119862A)


J119871(119883P119862A)



120572 120572 isin



119862A


(119883P119862A) P

119862A = and J119877(119883P119862A)

P119862A =




A where


120572ge

120575120572(119864)

2119862120572


120572(119864) = sup 119901

120572(119906 119908) 119906 119908 isin119864

(14)

If J119862A = 119869


119869120572 119883


119869120572(119906 119908) =

119901120572(119906 119908) 119894119891 119864 cap 119906 119908 = 119906 119908

120583120572

119894119891 119864 cap 119906 119908 = 119906 119908

(15)


119862A


1198691205720

(1199060 1199080) gt 1198621205720

[1198691205720

(1199060 V0) + 1198691205720

(V0 1199080)] (16)



12057201205831205720 Therefore by (14) 119901

1205720(1199060 1199080) gt

211986212057201205831205720



gt

1198621205720[119901

1205720(1199060 V0) + 120583

1205720] ge 119862

12057201205831205720


gt

1198621205720[120583

1205720+120583

1205720] = 2119862



gt

1198621205720[120583

1205720+ 119901

1205720(V0 1199080)] ge 119862

12057201205831205720


gt

1198621205720[120583

1205720+120583

1205720] = 2119862



gt

1198621205720[120583

1205720+ 120583

1205720] = 2119862



119906V119908isin119883 119869120572(119906 119908) le 119862

120572[119869

120572(119906 V) + 119869

120572(V


119898 119898 isin N) and (119908

119898



1198980=1198980(120572)isinNforall119898ge1198980

119869120572(119908

119898 119906

119898) lt 120576 (17)

Denoting 1198981015840


119898ge119898

1015840 119864 cap 119908119898 119906

119898 = 119908

119898 119906

119898



119898ge1198981015840 119901

120572(119908

119898 119906

119898) = 119869

120572(119908

119898 119906

119898) lt 120576


119898 isin N) and(119908




119898 119898 isin N)

and (119908119898



119862A isin

J119871(119883P119862A)

cap J119877(119883P119862A)

holds




119862A If P119862A


119883 that is

forall119906119908isin119883

119906 =119908


120572(119906 119908) gt 0or 119869

120572(119908 119906) gt 0

(18)

holds


120572(1199060 1199080) = 0 and 119869

120572(1199080 1199060) = 0 Then (J1) implies


120572(1199060 1199060) le 119862

120572[119869

120572(1199060 1199080)+119869

120572(1199080 1199060)] = 0 or equiv-


120572(1199060 1199060) = 119869

120572(1199080 1199060) = 0 and forall

120572isinA 119869120572(1199060

1199060) = 119869120572(1199060 1199080) = 0 Assuming that 119906

119898= 1199060 and 119908

119898= 1199080


119898rarrinfinsup

119899gt119898119869120572(119906

119898 119906

119899) =

lim119898rarrinfin

119869120572(119908

119898 119906



sup119899gt119898

119869120572(119906

119899

119906119898) = lim

119898rarrinfin119869120572(119906

119898 119908






119898rarrinfin119901120572(119908

119898 119906

119898) = lim

119898rarrinfin119901120572(119906

119898 119908

119898) = 0 or


120572(1199080 1199060) = 119901

120572(1199060 1199080) =








119898 119898 isin N) sub 119883 is left


119898rarrinfinsup

119899gt119898119869120572(119906

119898

119906119899) = 0 (forall


sup119899gt119898

119869120572(119906

119899 119906

119898) = 0)

(B) Let 119906 isin 119883 and let (119906119898





(119906119898 119898isinN)= ) where


= 119909


119898rarrinfin

119869120572(119909 119906

119898) = 0



119898rarrinfin

119869120572(119906

119898 119909) = 0)

(19)


119898 isin N) sub


(119906119898 119898isinN)=







= )) then (119883P119862A) is











sub



119898 isin N) of (119906119898

119898 isin N)



119898



) is a singleton







(A) Given 1199080



119898


119898isin0cupN 119908119898+1

isin 119879(119908119898





119898rarrinfin

sup119899gt119898

119869120572(119908

119898

119908119899

) = 0


119898rarrinfin

sup119899gt119898

119869120572(119908

119899

119908119898

) = 0)

(20)



isin

119883




119862A)



holds




[119896]



119898 119898 isin N) in 119879

[119896]





119898 119898 isin N) and (119906


forall119898isinN V

119898isin 119879

[119896]





119879[119896]

(119909) (119909 isin 119879[119896]

(119909))







119862A Let120582 = 120582




119909isin119883forall119881isin2119883 119869

120572(119909 119881) = inf 119869

120572(119909 V) V isin119881

and 119869120572(119881 119909) = inf 119869

120572(V 119909) V isin119881

(21)


(A) LetJ119862A isin J119871

(119883P119862A) If


119880119882isin2119883 119863119871minusJ119862A

12119883120572 (119880119882)


119869120572(119906119882) sup

119908isin119882

119869120572(119880 119908)


119880119882isin2119883 119863119871minusJ119862A

22119883120572 (119880119882)


119869120572(119906119882) sup

119908isin119882

119869120572(119908 119880)


119880119882isin2119883 119863119871minusJ119862A

32119883120572 (119880119882) = sup119906isin119880

119869120572(119906119882)

(22)


1205782119883 = 119863119871minusJ119862A





119909119910isin119883119862

120572sdot 119863

119871minusJ119862A

1205782119883120572 (119879 (119909) 119879 (119910))

le 120582120572119869120572(119909 119910)

(23)


1205782119883 120582)-quasi-119888119900119899119905119903119886119888119905119894119900119899on 119883

(B) LetJ119862A isin J119877

(119883P119862A) If


119880119882isin2119883 119863119877minusJ119862A

12119883120572 (119880119882)


119869120572(119906119882) sup

119908isin119882

119869120572(119880 119908)


119880119882isin2119883 119863119877minusJ119862A

22119883120572 (119880119882)


119869120572(119906119882) sup

119908isin119882

119869120572(119908 119880)


119880119882isin2119883 119863119877minusJ119862A

32119883120572 (119880119882) = sup119906isin119880

119869120572(119906119882)

(24)


1205782119883 = 119863119877minusJ119862A





119909119910isin119883119862

120572sdot 119863

119877minusJ119862A

1205782119883120572 (119879 (119909) 119879 (119910))

le 120582120572119869120572(119909 119910)

(25)




1205782119883 and D119877minusJ119862A




1205782119883



119877minusJ119862A






120572120572isinA isin [0 1)A




119871minusJ119862A


(D119877minusJ119862A





A



120572(119909 119910) lt 119869

120572(119909 119879 (119909)) + 120573

120572 (26)


120572(119910 119909) lt 119869

120572(119879 (119909) 119909) + 120573

120572) (27)


119898



119898+1isin 119879(119908

119898




[119896]

) is left (right)P




119898isin0cupN 119908119898+1

isin 119879(119908119898


) such that (119908119898






0isin 119883 and (119883 119879

[119896]




120572 119883


choose

119903 = 119903120572120572isinA

isin (0infin)A (28)


such that


120572(119908

0 119879 (119908

0)) lt(1minus

120582120572

119862120572

) 119903120572 (29)

Put


(0)120572

=(1minus120582120572

119862120572

) 119903120572minus 119869

120572(119908

0 119879 (119908

0)) (30)


= 120573(0)120572

120572isinA isin


1isin 119879(119908

0) such that


120572(119908

0 119908

1) lt 119869

120572(119908

0 119879 (119908

0)) + 120573

(0)120572

(31)



120572(119908

0 119908

1) lt(1minus

120582120572

119862120572

) 119903120572 (32)

Put now


(1)120572

=(120582120572

119862120572

)[(1minus120582120572

119862120572

) 119903120572minus 119869

120572(119908

0 119908

1)] (33)


(1)120572



2isin 119879(119908

1) such that


120572(119908

1 119908

2) lt 119869

120572(119908

1 119879 (119908

1)) + 120573

(1)120572

(34)

Observe that


120572(119908

1 119908

2) lt(

120582120572

119862120572

)(1minus120582120572

119862120572

) 119903120572 (35)



120572(119908

1 119908

2) lt 119869

120572(119908

1 119879 (119908

1)) + 120573

(1)120572

le sup 119869120572(119906 119879 (119908

1)) 119906 isin119879 (119908

0) + 120573

(1)120572

le119863119871minusJ119862A

1205782119883120572 (119879 (1199080) 119879 (119908

1)) + 120573

(1)120572

le(120582120572

119862120572

) 119869120572(119908

0 119908

1) + 120573

(1)120572

=(120582120572

119862120572

)(1minus120582120572

119862120572

) 119903120572

(36)



(2)120572

=(120582120572

119862120572

)[(120582120572

119862120572

)(1minus120582120572

119862120572

) 119903120572minus 119869

120572(119908

1 119908

2)]

(37)

Then in view of (35) 120573(2)= 120573

(2)120572



3isin

119879(1199082) such that


120572(119908

2 119908

3) lt 119869

120572(119908

2 119879 (119908

2)) + 120573

(2)120572

(38)



120572(119908

2 119908

3) lt(

120582120572

119862120572

)

2

(1minus120582120572

119862120572

) 119903120572 (39)



120572(119908

2 119908

3) lt 119869

120572(119908

2 119879 (119908

2)) + 120573

(2)120572

⩽ sup119906isin119879(119908

1)

119869120572(119906 119879 (119908

2)) + 120573

(2)120572

le119863119871minusJ119862A

1205782119883120572 (119879 (1199081) 119879 (119908

2)) + 120573

(2)120572

le(120582120572

119862120572

) 119869120572(119908

1 119908

2) + 120573

(2)120572

=(120582120572

119862120572

)

2

(1minus120582120572

119862120572

) 119903120572

(40)





119898+1isin119879 (119908

119898

) (41)


(119898)

=

120573(119898)

120572120572isinA

where


119898isinN 120573(119898)

120572=(

120582120572

119862120572

)

sdot [(120582120572

119862120572

)

119898minus1(1minus

120582120572

119862120572

) 119903120572minus 119869

120572(119908

119898minus1 119908

119898

)]

(42)


(119898)

isin (0infin)A


119898isinN 119869120572(119908

119898

119908119898+1

) lt 119869120572(119908

119898

119879 (119908119898

))

+ 120573(119898)

120572

(43)


119898isinN 119869120572(119908

119898

119908119898+1

)

lt(120582120572

119862120572

)

119898

(1minus120582120572

119862120572

) 119903120572

(44)



forall120572isinA

119869120572(119908

119898

119908119899

) ⩽119862120572119869120572(119908

119898

119908119898+1

)

+1198622120572119869120572(119908

119898+1 119908


+ 119862119899minus119898minus1120572

119869120572(119908

119899minus2 119908

119899minus1) +119862

119899minus119898minus1120572

119869120572(119908

119899minus1 119908

119899

)

=


119895=0119862119895+1120572

119869120572(119908

119898+119895

119908119898+119895+1

)

+119862119899minus119898minus1120572

119869120572(119908

119899minus1 119908

119899

)

(45)


119869120572(119908

119898

119908119899

) lt (1minus120582120572

119862120572

)

sdot 119903120572

[

[


119895=0119862119895+1120572

(120582120572

119862120572

)

119898+119895

+119862119899minus119898minus1120572

(120582120572

119862120572

)

119899minus2]

]

= (1minus120582120572

119862120572

)

sdot 119903120572

[

[

119862120572(

120582120572

119862120572

)

119898 119899minus119898minus2sum

119895=0120582119895

120572+(

119862120572

1205822120572

)120582119899

120572

119862119898

120572

]

]

(46)



119898isin0cupN 119908119898+1

isin119879 (119908119898

) (47)


120572le 120582

119898

120572


119898rarrinfin

sup119899gt119898

119869120572(119908

119898

119908119899


sup119899gt119898

(1minus120582120572

119862120572

)

sdot 119903120572[119862

120572(

120582120572

119862120572

)

119898

(1 minus 120582120572)minus1

+(119862120572

1205822120572

)120582119899

120572

119862119898

120572

]


(1minus120582120572

119862120572

) 119903120572[119862

120572(

120582120572

119862120572

)

119898

(1minus120582120572)minus1

+(119862120572

1205822120572

)(120582120572

119862120572

)

119898

]= 0

(48)


0isin 119883 by



119898rarrinfin

119869120572(119908 119908

119898

) = 0 (49)


= 119908119898 and 119908

119898= 119908 for 119898 isin N by



119898 isin N) and (119908119898



119898rarrinfin

119901120572(119908 119908

119898


119901120572(119908

119898 119906

119898) = 0 (50)



[119896]



119908(119898+1)119896

isin 119879[119896]

(119908119898119896


=

119908119898minus1+119896

119898 isin N) we see that (119909119898

119898 isin N) sub 119879[119896]

(119883)LIM119871minusP119862A


119898=

119908(119898+1)119896

119898 isin N) sub 119879[119896]

(119883) and (119906119898

= 119908119898119896

119898 isin N) sub

119879[119896]


119898isin 119879

[119896]


of (119909119898








[119896] is left P119862A-



such that 119908 isin 119879[119896]

(119908)



119862A)









119862A isin J119871(119883P119862A)


-


= 119863119871minusJ119862A120578119883120572

1198832



119906119908isin119883119863

119871minusJ119862A1119883120572 (119906 119908)

=max 119869120572(119906 119908) 119869

120572(119908 119906)


119906119908isin119883119863

119871minusJ119862A2119883120572 (119906 119908) = 119869

120572(119906 119908)

(51)





119909119910isin119883119862

120572sdot 119863

119871minusJ119862A120578119883120572

(119879 (119909) 119879 (119910))

le 120582120572119869120572(119909 119910)

(52)

(B) If J119862A isin J119877


D119877minusJ119862A119883120578


= 119863119877minusJ119862A120578119883120572

1198832

rarr



119906119908isin119883119863

119877minusJ119862A1119883120572 (119906 119908)

=max 119869120572(119906 119908) 119869

120572(119908 119906)


119906119908isin119883119863

119877minusJ119862A2119883120572 (119906 119908) = 119869

120572(119906 119908)

(53)




119909119910isin119883119862

120572sdot 119863

119877minusJ119862A120578119883120572

(119879 (119909) 119879 (119910))

le 120582120572119869120572(119909 119910)

(54)






(A) Given 1199080



119898

= 119879[119898]

(1199080)


(119908119898119898isin0cupN) = )

whenever


119898rarrinfin

sup119899gt119898

119869120572(119908

119898

119908119899

) = 0


119898rarrinfin

sup119899gt119898

119869120572(119908

119899

119908119898

) = 0)

(55)



isin 119883









[119896]

) is


(119909119898

119898 isin N) in 119879[119896]





119898 119898 isin N) and (119906


forall119898isinN V

119898= 119879

[119896]





119879[119896]

(119909) (119909 = 119879[119896]

(119909))


119862A)



120572120572isinA isin [0 1)A




119871minusJ119862A119883120578


119877minusJ119862A119883120578




(119908119898

= 119879[119898]



[119896]

) is left(right)P




119898

= 119879[119898]




120572(V 119879 (V)) = 119869

120572(119879 (V) V) = 0 (56)




) is left (right)P


Fix (119879[119896]

) = Fix (119879) = 119908 (57)


= 119879[119898]




120572(119908 119908) = 0 (58)




[119896]

(V) = 119879[2119896]

(V) 119879(V) =

119879[2119896]


0 lt 1198691205720

(V 119879 (V)) = 1198691205720

(119879[2119896]

(V) 119879[2119896](119879 (V)))

le 119863119871minusJ119862A1205781198831205720

(119879[2119896]

(V) 119879[2119896](119879 (V)))


le (1205821205720

1198621205720

)1198691205720

(119879[2119896minus1]

(V) 119879[2119896minus1](119879 (V)))

le (1205821205720

1198621205720

) sdot119863119871minusJ119862A1205781198831205720

(119879[2119896minus1]

(V) 119879[2119896minus1](119879 (V)))

le (1205821205720

1198621205720

)

2

1198691205720

(119879[2119896minus2]


le (1205821205720

1198621205720

)

2119896

1198691205720

(V 119879 (V)) lt 1198691205720

(V 119879 (V))

(59)



120572(V 119879 (V)) = 0 (60)




119879[119896]

(V) = 119879[2119896]


0 lt 1198691205720

(119879 (V) V) = 1198691205720

(119879[119896+1]

(V) 119879[2119896](V))

le

119896minus2sum

119898=1119862119898

12057201198691205720

(119879[119896+119898]

(V) 119879[119896+119898+1](V))

+119862119896minus21205720

1198691205720

(119879[2119896minus1]

(V) 119879[2119896](V))

le

119896minus2sum

119898=1119862119898

1205720sdot 119863

119871minusJ119862A1205781198831205720

(119879[119896+119898]

(V) 119879[119896+119898+1](V))

+119862119896minus21205720

sdot 119863119871minusJ119862A1205781198831205720

(119879[2119896minus1]

(V) 119879[2119896](V))

le

119896minus2sum

119898=1119862119898

1205720(

1205821205720

1198621205720

)

119896+119898

1198691205720

(V 119879 (V))

+119862119896minus21205720

(1205821205720

1198621205720

)

2119896minus1

1198691205720

(V 119879 (V)) = 0

(61)



120572(119879 (V) V) = 0 (62)






)


120572 120572 isin A is


120572(119879(V) V) gt



120572(119879(V) V) gt 0 or 119869

120572(V 119879(V)) gt 0 However by


120572(V V) = 0 In fact


120572(V V) le 119862

120572[119869

120572(V 119879(V))+119869

120572(119879(V) V)] = 0




1199011205720(119906 119908) gt 0 or 119901

1205720(119908 119906) gt 0 By applying


1198691205720(119906 119908) gt 0 or 119869

1205720(119908 119906) gt


exist1205720isinA

[1198691205720

(119906 119908) gt 0 1198691205720

(119906 119908)

= 1198691205720

(119879 (119906) 119879 (119908)) le119863119871minusJ119862A1205781198831205720

(119879 (119906) 119879 (119908))

le(1205821205720

1198621205720

)1198691205720

(119906 119908) lt 1198691205720

(119906 119908)] or [1198691205720

(119908 119906)

gt 0 1198691205720

(119908 119906) = 1198691205720

(119879 (119908) 119879 (119906))

le119863119871minusJ119862A1205781198831205720

(119879 (119908) 119879 (119906)) le(1205821205720

1198621205720

)1198691205720

(119908 119906)

lt 1198691205720

(119908 119906)]

(64)




rarr [0infin)

be of the form

119901 (119906 119908)

=

0 if 119906 ge 119908 119906 119908 cap (0 6) = 119906 119908

(119908 minus 119906)4 if 119906 lt 119908 119906 119908 cap (0 6) = 119906 119908

120574 if 119906 119908 cap (0 6) = 119906 119908

(65)


space In fact





8(119908 minus V)4 = 8[119901(119906 V) + 119901(V 119908)]


(119908 minus 119906)4 and 119891(V0) = min



and V0 = (119906 + 119908)2


119906119908isin(06)119906lt119908(119908 minus 119906)4

=


119906leVle1199088[119901(119906 V) + 119901(V 119908)] =

sup119906119908isin(06)119906lt119908min


(6 minus 3)4] = 8[81 + 81] = 1296




4lt

8(V minus 119906)4= 8[119901(119906 V) + 119901(V 119908)]


1296 le 8[119901(119906 V) + 119901(V 119908)] = 8[120574 + 120574]


8120574 le 8[119901(119906 V) + 119901(V 119908)]



119901(119906119908) = 119901(119908 119906) does not hold




119898= 3





(119906119898 119898isinN)= [0 3]



119901 (119906 119908) =

0 if 119860 cap 119906 119908 = 119906 119908

120574 if 119860 cap 119906 119908 = 119906 119908

(67)









119901(119906 119906) = 0 does not hold(3)P


(119906119898 119898isinN)= LIM119877minusP11

(119906119898 119898isinN)= 119860 for





119901 (119906 119908) =

0 if 119906 ge 119908

(119908 minus 119906)3 if 119906 lt 119908

(68)






4119901(V 119908) = 4[119901(119906 V) + 119901(V 119908)]


119891(V0) = min119906⩽V⩽119908119891(V) = (119908 minus 119906)

3 where V0 = (119906 + 119908)2 is


4(119908 minus 119906)[1199082+ 119908119906 + 119906

2+ 3V2 minus 3V(119908 + 119906)]



3lt 4(V minus 119906)

3= 4119901(119906 V) = 4[119901(119906

V) + 119901(V 119908)]



119901(119906119908) = 119901(119908 119906) does not hold




(119906119898 119898isinN)= [2 6] and

LIM119877minusP41(119906119898 119898isinN)

= [1 2] for sequence (119906119898



1198832


119901 (119906 119908) =

0 if 119906 ge 119908

(119906 minus 119908)2 if 119906 lt 119908

(69)

Let

119864 = [0 3) cup (3 6] (70)


2rarr [0infin) is

of the form

119869 (119906 119908) =

119901 (119906 119908) if 119864 cap 119906 119908 = 119906 119908

120583 if 119864 cap 119906 119908 = 119906 119908

(71)


36 and 119869(6 0) = 0(2)J

21 = 119869 isin J119871(119883P21)

capJ119877(119883P21)

SeeTheorem 14




119901 (119906 119908)

=

0 if 119906 ge 119908 119906 119908 cap (0 6) = 119906 119908

(119908 minus 119906)4 if 119906 lt 119908 119906 119908 cap (0 6) = 119906 119908

120574 if 119906 119908 cap (0 6) = 119906 119908

(72)


119879 (119906) =

[1 2] if 119906 isin [0 3) cup (4 6]

(4 6) if 119906 isin [3 4] (73)


Let119864 = [0 3) cup (4 6] (74)


119869 (119906 119908) =

119901 (119906 119908) if 119864 cap 119906 119908 = 119906 119908

120574 if 119864 cap 119906 119908 = 119906 119908

(75)

(1) (119883P81) where P




(2) (119883 119879) is a (119863 = D119871minusJ8112119883 = D

119877minusJ8112119883 120582 isin [2048


119909119910isin1198838 sdot

119863(119879(119909) 119879(119910)) le 120582119869(119909 119910) where

119863 (119880119882) = maxsup119906isin119880

119869 (119906119882) sup119908isin119882

119869 (119880 119908)

119880119882 isin 2119883

(76)



119864 and sup119906isin119880

inf119908isin119880

119869(119906 119908) = sup119906isin119880

119869(119906 119906) = 119901(119906 119906) =

0 = 0 Thus 4119863(119879(119909) 119879(119910)) = 0 le 120582119869(119909 119910)



inf119908isin119882

119869(119906 119908) =

sup119906isin119880

inf119908isin119882

(119908 minus 119906)4 = sup

119906isin119880(4 minus 119906)

4= 81 and

sup119908isin119882

inf119906isin119880

119869(119906 119908) = sup119908isin119882

inf119906isin119880

(119908 minus 119906)4 =

sup119908isin119882





119869(119906 119908) = sup119906isin119880

inf119908isin119882

119901(119906 119908) =

sup119908isin119882

inf119906isin119880

119869(119906 119908) = sup119908isin119882

inf119906isin119880

119901(119906 119908) = 0Therefore 8119863(119879(119909) 119879(119910)) = 0 le 120582119869(119909 119910)




exist119910isin119879(119909)

119869(119909




120573isin(0infin)119869(1199090

1199100) lt 119869(1199090 119879(1199090)) + 120573



120573isin(0infin)119869(1199090 1199100) lt 119869(1199090 119879(1199090)) + 120573


120573isin(0infin)119869(1199090 1199100) lt

119869(1199090 119879(1199090)) + 120573



119869(1199090 1199100) lt 119869(1199090 119879(1199090)) +

120573


120573isin(0infin)119869(1199090 1199100) lt 119869(1199090 119879(1199090)) + 120573


1199080isin 119883 In fact if1199080

isin 119883 and (119908119898


119898+1isin 119879(119908

119898


sup119899gt119898

119869(119908119898

119908119899

) =


sup119899gt119898

119869(119908119899

119908119898

) = 0) then forall119898ge2 119908

119898



119901(119908119908119898

) =


119898rarrinfin119901(119908

119898



119869(119908 119908119898

) =


119898rarrinfin119869(119908

119898





(119909119898 119898isinN)= (LIM119877minusP81

(119909119898 119898isinN)= ))


119898 isin N) and (119906119898

119898 isin N)


119898isin 119879(119906

119898) Then forall

119898ge2 119909119898

isin [1 2] 2 isin





following


0isin 119883




0isin 119883





119909119910isin1198838119863(119879(119909)


119901(119906119882) sup



119909119910isin1198838119863(119879(119909) 119879(119910)) le 1205820119901(119909 119910) Letting 1199090 = 2


[1 2] = 119880 119879(1199100) = (4 6) = 119882 sup119906isin[12]119901(119906 (4 6)) =

sup119906isin[12](4 minus 119906)

4= 34 = 81 and sup

119908isin(46)119901([1 2] 119908) =




119862A)


119862A whereJ



119862A)whenJ119862A = P


119862A





119860 = 1198601 cup1198602 1198601 = (0 2] 1198602 = [4 6) (77)

Let 119901 1198832


119901 (119906 119908) =

0 if 119860 cap 119906 119908 = 119906 119908

120574 if 119860 cap 119906 119908 = 119906 119908

(78)

and letJ11 = P


119879 (119906) =

1198602 for 119906 isin (0 3)

119860 for 119906 = 3

1198601 for 119906 isin (3 6)

(79)


Section 11(2) (119883 119879) is a (D



119909119910isin119883119863

119871minusP1112119883 (119879(119909) 119879(119910)) le 120582119901(119909


and maxsup119906isin119879(119909)

119901(119906 119879(119910)) sup119908isin119879(119910)



exist119910isin119879(119909)



119901 (1199090 1199100) = 119901 (1199090 119879 (1199090))

=

0 if 1199090 isin 1198601

120574 for 1199090 isin (0 3) 1198601

(80)

Therefore 119901(1199090 1199100) lt 119901(1199090 119879(1199090)) + 120573



119901 (1199090 1199100) = 119901 (1199090 119879 (1199090))

=

0 if 1199090 isin 1198602

120574 for 1199090 isin (3 6) 1198602

(81)







sup119899gt119898

119901(119908119898

119908119899


lim119898rarrinfin

119901(119908119908119898


(79)forall119898⩾1 119908

119898

isin 119879(119908119898minus1



(119908119898119898isin0cupN)




119898 isin N) sub 119879[2]

(119883) = 119860



119898 isin N) and (119906119898


119898isin 119879(119906


exist1198980isinN

forall119898⩾1198980

119909119898



and Fix(119879[2]) = 119860

(6) For (119883P11)P11 = 119901 J

11 = P11 and



isin

119883 1199081isin 119879(119908

0) and 119908

2= 119906 isin 119879(119908


and forall119898⩾3 119908

119898

= 119906 then sequence (119908119898




1198832


119901 (119906 119908) =

0 if 119906 ge 119908

(119906 minus 119908)2 if 119906 lt 119908

(82)


119879 (119906) = ([0 3) cup (3 6]) 119906 for 119906 isin [0 6] (83)

Let

119864 = [0 3) cup (3 6] (84)


2rarr [0infin) is

of the form

119869 (119906 119908) =

119901 (119906 119908) if 119864 cap 119906 119908 = 119906 119908

120583 if 119864 cap 119906 119908 = 119906 119908

(85)


119869(0 6) = 36 and 119869(6 0) = 0(2)J

21 = 119869 isin J119871(119883P21)

capJ119877(119883P21)

SeeTheorem 14




[0 1) and

119863(119880119882) = maxsup119906isin119880

119869 (119906119882) sup119908isin119882

119869 (119880 119908)

119880119882 isin 2119883

(86)




119906

cup 119882119906and

inf119908isin119882

119869 (119906 119908)

le

inf119908isin119882119906119902 (119906 119908) = 0 if 119882

119906

= 119908 isin 119882 119906 ge 119908 =

inf119908isin119882119906

(119906 minus 119908)2= 0 if 119882

119906= 119908 isin 119882 119906 lt 119908 =

(87)


cup 119880119908and

inf119906isin119880

119869 (119906 119908)

le

inf119906isin119880119908119902 (119906 119908) = 0 if 119880

119906

= 119906 isin 119880 119906 ge 119908 =

inf119906isin119880119908

(119906 minus 119908)2= 0 if 119880

119906= 119906 isin 119880 119906 lt 119908 =

(88)

By (86) 2119863(119879(119909) 119879(119910)) = 0 le 120582119869(119909 119910)

Case 2 If 119909 = 119910 = 3 then 119869(119909 119910) = 120583 and 119879(119909) =


2119863(119880119880) = 0 le 120582119869(119909 119910)



inf119908isin119882

119869(119906 119908) =


119869(119906 119908) =


inf119906isin119880


119908

cup 119880119908and

inf119906isin119880

119869 (119906 119908)

le

inf119906isin119880119908119902 (119906 119908) = 0 if 119880

119906

= 119906 isin 119880 119906 ge 119908 =

inf119906isin119880119908

(119906 minus 119908)2= 0 if 119880

119906= 119906 isin 119880 119906 lt 119908 =

(89)

Thus 2119863(119879(119909) 119879(119910)) = 0 ⩽ 120582119869(119909 119910)


(3 6]) 119910 = 119882 sub 119864 and sup119906isin119880

inf119908isin119882

119869(119906 119908) = 0 sincefor 119906 isin 119880

inf119908isin119882

119869 (119906 119908)

le

inf119908isin119882119906119902 (119906 119908) = 0 if 119882

119906

= 119908 isin 119882 119906 ge 119908 =

inf119908isin119882119906

(119906 minus 119908)2= 0 if 119882

119906= 119908 isin 119882 119906 lt 119908 =

(90)


inf119906isin119880

119869(119906 119908) = 0 since inf119906isin119880

119869(119906 119908) = 119869(119908119908) = 0 for 119908 isin 119882 Thus 2119863(119879(119909) 119879(119910)) = 0 le 120582119869(119909 119910)



exist119910isin119879(119909)

119869(119909




inf119908isin119882

119869(1199090 119908) = 0 since

inf119908isin119882

119869 (1199090 119908)

le

inf119908isin1198821199090119902 (1199090 119908) = 0 if 119882

1199090 = 119908 isin 119882 1199090 ge 119908 =

inf119908isin1198821199090

(1199090 minus 119908)2= 0 if 119882

1199090= 119908 isin 119882 1199090 lt 119908 =

(91)



12)

then 119869(1199090 1199100) lt 119869(1199090 119879(1199090)) + 120574



119869(1199090 1199100) lt 119869(1199090 119879(1199090)) + 120574




119869(1199090 119879(1199090)) + 120574


1199080


119898



sup119899gt119898

119869(119908119898

119908119899


lim119898rarrinfin

119869(119908 119908119898



1isin 119879(119908

0) = ([0 3) cup

(3 6]) 1199080 and forall

119898ge2 119908119898

isin 119879(119908119898minus1


(119908119898119898isin0cupN)

Case 2 If 1199080= 3 then 119908

1isin 119879(119908

0) = [0 3) cup (3 6] 1199082

isin

119879(1199081) = ([0 3) cup (3 6]) 119908

1 and forall

119898⩾3 119908119898

isin 119879(119908119898minus1

) sub




0isin 119883 and



119898rarrinfinsup

119899gt119898119869(119908

119898

119908119899

) =

0 of (119908119898


[2]) is a left P


119898 119898 isin N) sub 119879

[2](119883) =



119898 isin N) and (119906119898

119898 isin N)


119898isin 119879(119906


exist1198980isinN

forall119898ge1198980

119909119898


(119909119898 119898isinN)and 6 isin 119879

[2](6)

(7) ForP21 = 119901J



that 6 isin 119879[2]


0isin 119883






11 = 119901


119879 (119906) =

4 for 119906 isin (0 3)

2 for 119906 isin [3 6) (92)



119909119910isin119883119863

119871minusP111119883 (119879(119909) 119879(119910)) =

max119901(119879(119909) 119879(119910)) 119901(119879(119910) 119879(119909)) le 120582119901(119909 119910) and Fix(119879) =






119898+1= 119879(119908

119898


sup119899gt119898

119901(119908119898

119908119899

) =


119898



(119908119898119898isin0cupN)




119898 isin N) sub

119879[2]



119898 isin N) and (119906119898

119898 isin N)


119898= 119879

[2](119906

119898) then by (77) (78) and (92)



(2) isin

LIM119871minusP11(119909119898 119898isinN)

and 4 = 119879[2]





119883119860 then 119901(119906 119908) = 119901(119908 119906) = 120574 gt 0(6) For P

11 = 119901 (119883 119879) and J11 = P





119889


119889



(119883 119889) rarr (CB(119883)119867119889











References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
























119862A = 119901120572

1198832

rarr



119906V119908isin119883 119901120572(119906 119908) le119862

120572[119901

120572(119906 V) + 119901

120572(V 119908)] (4)





forall119906119908isin119883

119906 =119908


120572(119906 119908) gt 0or119901

120572(119908 119906) gt 0

(5)


120572isinA

forall119906119908isin119883

119901minus1120572

(119906 119908) = 119901120572(119908 119906) then forall


minus1120572

(119906

119908) le 119862120572[119901

minus1120572

(119906 V) + 119901minus1120572


119862A) Pminus1119862A = 119901

minus1120572

1198832

rarr [0infin)



ces 119901120572

1198832



119901120572(119906 119906) = 0 or


119906119908isin119883119901

120572(119906 119908) = 119901

120572(119908 119906) or forall


120572(119906

119908) le 119901120572(119906 V) + 119901


Section 10



A


120572 119883

2rarr [0infin)



119906V119908isin119883 119897120572(119906 119908)

le119862120572max 119897

120572(119906 V) 119897

120572(V 119908)

(6)




120572 119883

2rarr [0infin)



119906V119908isin119883 119904120572(119906 119908) le119862

120572[119904

120572(119906 V) + 119904

120572(V 119908)]

minus 119904120572(V V)

(7)






119862A)






(A) The family J119862A = 119869


119869120572

1198832


119862A if


119906V119908isin119883 119869120572(119906 119908) le 119862

120572[119869

120572(119906 V) + 119869

120572(V 119908)]

and furthermore



119898 isin N) and (119908119898



119898rarrinfin

sup119899gt119898

119869120572(119906

119898 119906

119899) = 0 (8)


119898rarrinfin

sup119899gt119898

119869120572(119906

119899 119906

119898) = 0) (9)


119898rarrinfin

119869120572(119908

119898 119906

119898) = 0 (10)


119898rarrinfin

119869120572(119906

119898 119908

119898) = 0) (11)

the following holds


119898rarrinfin

119901120572(119908

119898 119906

119898) = 0 (12)


119898rarrinfin

119901120572(119906

119898 119908

119898) = 0) (13)

(B) J119871(119883P119862A)

(J119877(119883P119862A)




J119871(119883P119862A)

cap J119877(119883P119862A)


J119871(119883P119862A)



120572 120572 isin



119862A


(119883P119862A) P

119862A = and J119877(119883P119862A)

P119862A =




A where


120572ge

120575120572(119864)

2119862120572


120572(119864) = sup 119901

120572(119906 119908) 119906 119908 isin119864

(14)

If J119862A = 119869


119869120572 119883


119869120572(119906 119908) =

119901120572(119906 119908) 119894119891 119864 cap 119906 119908 = 119906 119908

120583120572

119894119891 119864 cap 119906 119908 = 119906 119908

(15)


119862A


1198691205720

(1199060 1199080) gt 1198621205720

[1198691205720

(1199060 V0) + 1198691205720

(V0 1199080)] (16)



12057201205831205720 Therefore by (14) 119901

1205720(1199060 1199080) gt

211986212057201205831205720



gt

1198621205720[119901

1205720(1199060 V0) + 120583

1205720] ge 119862

12057201205831205720


gt

1198621205720[120583

1205720+120583

1205720] = 2119862



gt

1198621205720[120583

1205720+ 119901

1205720(V0 1199080)] ge 119862

12057201205831205720


gt

1198621205720[120583

1205720+120583

1205720] = 2119862



gt

1198621205720[120583

1205720+ 120583

1205720] = 2119862



119906V119908isin119883 119869120572(119906 119908) le 119862

120572[119869

120572(119906 V) + 119869

120572(V


119898 119898 isin N) and (119908

119898



1198980=1198980(120572)isinNforall119898ge1198980

119869120572(119908

119898 119906

119898) lt 120576 (17)

Denoting 1198981015840


119898ge119898

1015840 119864 cap 119908119898 119906

119898 = 119908

119898 119906

119898



119898ge1198981015840 119901

120572(119908

119898 119906

119898) = 119869

120572(119908

119898 119906

119898) lt 120576


119898 isin N) and(119908




119898 119898 isin N)

and (119908119898



119862A isin

J119871(119883P119862A)

cap J119877(119883P119862A)

holds




119862A If P119862A


119883 that is

forall119906119908isin119883

119906 =119908


120572(119906 119908) gt 0or 119869

120572(119908 119906) gt 0

(18)

holds


120572(1199060 1199080) = 0 and 119869

120572(1199080 1199060) = 0 Then (J1) implies


120572(1199060 1199060) le 119862

120572[119869

120572(1199060 1199080)+119869

120572(1199080 1199060)] = 0 or equiv-


120572(1199060 1199060) = 119869

120572(1199080 1199060) = 0 and forall

120572isinA 119869120572(1199060

1199060) = 119869120572(1199060 1199080) = 0 Assuming that 119906

119898= 1199060 and 119908

119898= 1199080


119898rarrinfinsup

119899gt119898119869120572(119906

119898 119906

119899) =

lim119898rarrinfin

119869120572(119908

119898 119906



sup119899gt119898

119869120572(119906

119899

119906119898) = lim

119898rarrinfin119869120572(119906

119898 119908






119898rarrinfin119901120572(119908

119898 119906

119898) = lim

119898rarrinfin119901120572(119906

119898 119908

119898) = 0 or


120572(1199080 1199060) = 119901

120572(1199060 1199080) =








119898 119898 isin N) sub 119883 is left


119898rarrinfinsup

119899gt119898119869120572(119906

119898

119906119899) = 0 (forall


sup119899gt119898

119869120572(119906

119899 119906

119898) = 0)

(B) Let 119906 isin 119883 and let (119906119898





(119906119898 119898isinN)= ) where


= 119909


119898rarrinfin

119869120572(119909 119906

119898) = 0



119898rarrinfin

119869120572(119906

119898 119909) = 0)

(19)


119898 isin N) sub


(119906119898 119898isinN)=







= )) then (119883P119862A) is











sub



119898 isin N) of (119906119898

119898 isin N)



119898



) is a singleton







(A) Given 1199080



119898


119898isin0cupN 119908119898+1

isin 119879(119908119898





119898rarrinfin

sup119899gt119898

119869120572(119908

119898

119908119899

) = 0


119898rarrinfin

sup119899gt119898

119869120572(119908

119899

119908119898

) = 0)

(20)



isin

119883




119862A)



holds




[119896]



119898 119898 isin N) in 119879

[119896]





119898 119898 isin N) and (119906


forall119898isinN V

119898isin 119879

[119896]





119879[119896]

(119909) (119909 isin 119879[119896]

(119909))







119862A Let120582 = 120582




119909isin119883forall119881isin2119883 119869

120572(119909 119881) = inf 119869

120572(119909 V) V isin119881

and 119869120572(119881 119909) = inf 119869

120572(V 119909) V isin119881

(21)


(A) LetJ119862A isin J119871

(119883P119862A) If


119880119882isin2119883 119863119871minusJ119862A

12119883120572 (119880119882)


119869120572(119906119882) sup

119908isin119882

119869120572(119880 119908)


119880119882isin2119883 119863119871minusJ119862A

22119883120572 (119880119882)


119869120572(119906119882) sup

119908isin119882

119869120572(119908 119880)


119880119882isin2119883 119863119871minusJ119862A

32119883120572 (119880119882) = sup119906isin119880

119869120572(119906119882)

(22)


1205782119883 = 119863119871minusJ119862A





119909119910isin119883119862

120572sdot 119863

119871minusJ119862A

1205782119883120572 (119879 (119909) 119879 (119910))

le 120582120572119869120572(119909 119910)

(23)


1205782119883 120582)-quasi-119888119900119899119905119903119886119888119905119894119900119899on 119883

(B) LetJ119862A isin J119877

(119883P119862A) If


119880119882isin2119883 119863119877minusJ119862A

12119883120572 (119880119882)


119869120572(119906119882) sup

119908isin119882

119869120572(119880 119908)


119880119882isin2119883 119863119877minusJ119862A

22119883120572 (119880119882)


119869120572(119906119882) sup

119908isin119882

119869120572(119908 119880)


119880119882isin2119883 119863119877minusJ119862A

32119883120572 (119880119882) = sup119906isin119880

119869120572(119906119882)

(24)


1205782119883 = 119863119877minusJ119862A





119909119910isin119883119862

120572sdot 119863

119877minusJ119862A

1205782119883120572 (119879 (119909) 119879 (119910))

le 120582120572119869120572(119909 119910)

(25)




1205782119883 and D119877minusJ119862A




1205782119883



119877minusJ119862A






120572120572isinA isin [0 1)A




119871minusJ119862A


(D119877minusJ119862A





A



120572(119909 119910) lt 119869

120572(119909 119879 (119909)) + 120573

120572 (26)


120572(119910 119909) lt 119869

120572(119879 (119909) 119909) + 120573

120572) (27)


119898



119898+1isin 119879(119908

119898




[119896]

) is left (right)P




119898isin0cupN 119908119898+1

isin 119879(119908119898


) such that (119908119898






0isin 119883 and (119883 119879

[119896]




120572 119883


choose

119903 = 119903120572120572isinA

isin (0infin)A (28)


such that


120572(119908

0 119879 (119908

0)) lt(1minus

120582120572

119862120572

) 119903120572 (29)

Put


(0)120572

=(1minus120582120572

119862120572

) 119903120572minus 119869

120572(119908

0 119879 (119908

0)) (30)


= 120573(0)120572

120572isinA isin


1isin 119879(119908

0) such that


120572(119908

0 119908

1) lt 119869

120572(119908

0 119879 (119908

0)) + 120573

(0)120572

(31)



120572(119908

0 119908

1) lt(1minus

120582120572

119862120572

) 119903120572 (32)

Put now


(1)120572

=(120582120572

119862120572

)[(1minus120582120572

119862120572

) 119903120572minus 119869

120572(119908

0 119908

1)] (33)


(1)120572



2isin 119879(119908

1) such that


120572(119908

1 119908

2) lt 119869

120572(119908

1 119879 (119908

1)) + 120573

(1)120572

(34)

Observe that


120572(119908

1 119908

2) lt(

120582120572

119862120572

)(1minus120582120572

119862120572

) 119903120572 (35)



120572(119908

1 119908

2) lt 119869

120572(119908

1 119879 (119908

1)) + 120573

(1)120572

le sup 119869120572(119906 119879 (119908

1)) 119906 isin119879 (119908

0) + 120573

(1)120572

le119863119871minusJ119862A

1205782119883120572 (119879 (1199080) 119879 (119908

1)) + 120573

(1)120572

le(120582120572

119862120572

) 119869120572(119908

0 119908

1) + 120573

(1)120572

=(120582120572

119862120572

)(1minus120582120572

119862120572

) 119903120572

(36)



(2)120572

=(120582120572

119862120572

)[(120582120572

119862120572

)(1minus120582120572

119862120572

) 119903120572minus 119869

120572(119908

1 119908

2)]

(37)

Then in view of (35) 120573(2)= 120573

(2)120572



3isin

119879(1199082) such that


120572(119908

2 119908

3) lt 119869

120572(119908

2 119879 (119908

2)) + 120573

(2)120572

(38)



120572(119908

2 119908

3) lt(

120582120572

119862120572

)

2

(1minus120582120572

119862120572

) 119903120572 (39)



120572(119908

2 119908

3) lt 119869

120572(119908

2 119879 (119908

2)) + 120573

(2)120572

⩽ sup119906isin119879(119908

1)

119869120572(119906 119879 (119908

2)) + 120573

(2)120572

le119863119871minusJ119862A

1205782119883120572 (119879 (1199081) 119879 (119908

2)) + 120573

(2)120572

le(120582120572

119862120572

) 119869120572(119908

1 119908

2) + 120573

(2)120572

=(120582120572

119862120572

)

2

(1minus120582120572

119862120572

) 119903120572

(40)





119898+1isin119879 (119908

119898

) (41)


(119898)

=

120573(119898)

120572120572isinA

where


119898isinN 120573(119898)

120572=(

120582120572

119862120572

)

sdot [(120582120572

119862120572

)

119898minus1(1minus

120582120572

119862120572

) 119903120572minus 119869

120572(119908

119898minus1 119908

119898

)]

(42)


(119898)

isin (0infin)A


119898isinN 119869120572(119908

119898

119908119898+1

) lt 119869120572(119908

119898

119879 (119908119898

))

+ 120573(119898)

120572

(43)


119898isinN 119869120572(119908

119898

119908119898+1

)

lt(120582120572

119862120572

)

119898

(1minus120582120572

119862120572

) 119903120572

(44)



forall120572isinA

119869120572(119908

119898

119908119899

) ⩽119862120572119869120572(119908

119898

119908119898+1

)

+1198622120572119869120572(119908

119898+1 119908


+ 119862119899minus119898minus1120572

119869120572(119908

119899minus2 119908

119899minus1) +119862

119899minus119898minus1120572

119869120572(119908

119899minus1 119908

119899

)

=


119895=0119862119895+1120572

119869120572(119908

119898+119895

119908119898+119895+1

)

+119862119899minus119898minus1120572

119869120572(119908

119899minus1 119908

119899

)

(45)


119869120572(119908

119898

119908119899

) lt (1minus120582120572

119862120572

)

sdot 119903120572

[

[


119895=0119862119895+1120572

(120582120572

119862120572

)

119898+119895

+119862119899minus119898minus1120572

(120582120572

119862120572

)

119899minus2]

]

= (1minus120582120572

119862120572

)

sdot 119903120572

[

[

119862120572(

120582120572

119862120572

)

119898 119899minus119898minus2sum

119895=0120582119895

120572+(

119862120572

1205822120572

)120582119899

120572

119862119898

120572

]

]

(46)



119898isin0cupN 119908119898+1

isin119879 (119908119898

) (47)


120572le 120582

119898

120572


119898rarrinfin

sup119899gt119898

119869120572(119908

119898

119908119899


sup119899gt119898

(1minus120582120572

119862120572

)

sdot 119903120572[119862

120572(

120582120572

119862120572

)

119898

(1 minus 120582120572)minus1

+(119862120572

1205822120572

)120582119899

120572

119862119898

120572

]


(1minus120582120572

119862120572

) 119903120572[119862

120572(

120582120572

119862120572

)

119898

(1minus120582120572)minus1

+(119862120572

1205822120572

)(120582120572

119862120572

)

119898

]= 0

(48)


0isin 119883 by



119898rarrinfin

119869120572(119908 119908

119898

) = 0 (49)


= 119908119898 and 119908

119898= 119908 for 119898 isin N by



119898 isin N) and (119908119898



119898rarrinfin

119901120572(119908 119908

119898


119901120572(119908

119898 119906

119898) = 0 (50)



[119896]



119908(119898+1)119896

isin 119879[119896]

(119908119898119896


=

119908119898minus1+119896

119898 isin N) we see that (119909119898

119898 isin N) sub 119879[119896]

(119883)LIM119871minusP119862A


119898=

119908(119898+1)119896

119898 isin N) sub 119879[119896]

(119883) and (119906119898

= 119908119898119896

119898 isin N) sub

119879[119896]


119898isin 119879

[119896]


of (119909119898








[119896] is left P119862A-



such that 119908 isin 119879[119896]

(119908)



119862A)









119862A isin J119871(119883P119862A)


-


= 119863119871minusJ119862A120578119883120572

1198832



119906119908isin119883119863

119871minusJ119862A1119883120572 (119906 119908)

=max 119869120572(119906 119908) 119869

120572(119908 119906)


119906119908isin119883119863

119871minusJ119862A2119883120572 (119906 119908) = 119869

120572(119906 119908)

(51)





119909119910isin119883119862

120572sdot 119863

119871minusJ119862A120578119883120572

(119879 (119909) 119879 (119910))

le 120582120572119869120572(119909 119910)

(52)

(B) If J119862A isin J119877


D119877minusJ119862A119883120578


= 119863119877minusJ119862A120578119883120572

1198832

rarr



119906119908isin119883119863

119877minusJ119862A1119883120572 (119906 119908)

=max 119869120572(119906 119908) 119869

120572(119908 119906)


119906119908isin119883119863

119877minusJ119862A2119883120572 (119906 119908) = 119869

120572(119906 119908)

(53)




119909119910isin119883119862

120572sdot 119863

119877minusJ119862A120578119883120572

(119879 (119909) 119879 (119910))

le 120582120572119869120572(119909 119910)

(54)






(A) Given 1199080



119898

= 119879[119898]

(1199080)


(119908119898119898isin0cupN) = )

whenever


119898rarrinfin

sup119899gt119898

119869120572(119908

119898

119908119899

) = 0


119898rarrinfin

sup119899gt119898

119869120572(119908

119899

119908119898

) = 0)

(55)



isin 119883









[119896]

) is


(119909119898

119898 isin N) in 119879[119896]





119898 119898 isin N) and (119906


forall119898isinN V

119898= 119879

[119896]





119879[119896]

(119909) (119909 = 119879[119896]

(119909))


119862A)



120572120572isinA isin [0 1)A




119871minusJ119862A119883120578


119877minusJ119862A119883120578




(119908119898

= 119879[119898]



[119896]

) is left(right)P




119898

= 119879[119898]




120572(V 119879 (V)) = 119869

120572(119879 (V) V) = 0 (56)




) is left (right)P


Fix (119879[119896]

) = Fix (119879) = 119908 (57)


= 119879[119898]




120572(119908 119908) = 0 (58)




[119896]

(V) = 119879[2119896]

(V) 119879(V) =

119879[2119896]


0 lt 1198691205720

(V 119879 (V)) = 1198691205720

(119879[2119896]

(V) 119879[2119896](119879 (V)))

le 119863119871minusJ119862A1205781198831205720

(119879[2119896]

(V) 119879[2119896](119879 (V)))


le (1205821205720

1198621205720

)1198691205720

(119879[2119896minus1]

(V) 119879[2119896minus1](119879 (V)))

le (1205821205720

1198621205720

) sdot119863119871minusJ119862A1205781198831205720

(119879[2119896minus1]

(V) 119879[2119896minus1](119879 (V)))

le (1205821205720

1198621205720

)

2

1198691205720

(119879[2119896minus2]


le (1205821205720

1198621205720

)

2119896

1198691205720

(V 119879 (V)) lt 1198691205720

(V 119879 (V))

(59)



120572(V 119879 (V)) = 0 (60)




119879[119896]

(V) = 119879[2119896]


0 lt 1198691205720

(119879 (V) V) = 1198691205720

(119879[119896+1]

(V) 119879[2119896](V))

le

119896minus2sum

119898=1119862119898

12057201198691205720

(119879[119896+119898]

(V) 119879[119896+119898+1](V))

+119862119896minus21205720

1198691205720

(119879[2119896minus1]

(V) 119879[2119896](V))

le

119896minus2sum

119898=1119862119898

1205720sdot 119863

119871minusJ119862A1205781198831205720

(119879[119896+119898]

(V) 119879[119896+119898+1](V))

+119862119896minus21205720

sdot 119863119871minusJ119862A1205781198831205720

(119879[2119896minus1]

(V) 119879[2119896](V))

le

119896minus2sum

119898=1119862119898

1205720(

1205821205720

1198621205720

)

119896+119898

1198691205720

(V 119879 (V))

+119862119896minus21205720

(1205821205720

1198621205720

)

2119896minus1

1198691205720

(V 119879 (V)) = 0

(61)



120572(119879 (V) V) = 0 (62)






)


120572 120572 isin A is


120572(119879(V) V) gt



120572(119879(V) V) gt 0 or 119869

120572(V 119879(V)) gt 0 However by


120572(V V) = 0 In fact


120572(V V) le 119862

120572[119869

120572(V 119879(V))+119869

120572(119879(V) V)] = 0




1199011205720(119906 119908) gt 0 or 119901

1205720(119908 119906) gt 0 By applying


1198691205720(119906 119908) gt 0 or 119869

1205720(119908 119906) gt


exist1205720isinA

[1198691205720

(119906 119908) gt 0 1198691205720

(119906 119908)

= 1198691205720

(119879 (119906) 119879 (119908)) le119863119871minusJ119862A1205781198831205720

(119879 (119906) 119879 (119908))

le(1205821205720

1198621205720

)1198691205720

(119906 119908) lt 1198691205720

(119906 119908)] or [1198691205720

(119908 119906)

gt 0 1198691205720

(119908 119906) = 1198691205720

(119879 (119908) 119879 (119906))

le119863119871minusJ119862A1205781198831205720

(119879 (119908) 119879 (119906)) le(1205821205720

1198621205720

)1198691205720

(119908 119906)

lt 1198691205720

(119908 119906)]

(64)




rarr [0infin)

be of the form

119901 (119906 119908)

=

0 if 119906 ge 119908 119906 119908 cap (0 6) = 119906 119908

(119908 minus 119906)4 if 119906 lt 119908 119906 119908 cap (0 6) = 119906 119908

120574 if 119906 119908 cap (0 6) = 119906 119908

(65)


space In fact





8(119908 minus V)4 = 8[119901(119906 V) + 119901(V 119908)]


(119908 minus 119906)4 and 119891(V0) = min



and V0 = (119906 + 119908)2


119906119908isin(06)119906lt119908(119908 minus 119906)4

=


119906leVle1199088[119901(119906 V) + 119901(V 119908)] =

sup119906119908isin(06)119906lt119908min


(6 minus 3)4] = 8[81 + 81] = 1296




4lt

8(V minus 119906)4= 8[119901(119906 V) + 119901(V 119908)]


1296 le 8[119901(119906 V) + 119901(V 119908)] = 8[120574 + 120574]


8120574 le 8[119901(119906 V) + 119901(V 119908)]



119901(119906119908) = 119901(119908 119906) does not hold




119898= 3





(119906119898 119898isinN)= [0 3]



119901 (119906 119908) =

0 if 119860 cap 119906 119908 = 119906 119908

120574 if 119860 cap 119906 119908 = 119906 119908

(67)









119901(119906 119906) = 0 does not hold(3)P


(119906119898 119898isinN)= LIM119877minusP11

(119906119898 119898isinN)= 119860 for





119901 (119906 119908) =

0 if 119906 ge 119908

(119908 minus 119906)3 if 119906 lt 119908

(68)






4119901(V 119908) = 4[119901(119906 V) + 119901(V 119908)]


119891(V0) = min119906⩽V⩽119908119891(V) = (119908 minus 119906)

3 where V0 = (119906 + 119908)2 is


4(119908 minus 119906)[1199082+ 119908119906 + 119906

2+ 3V2 minus 3V(119908 + 119906)]



3lt 4(V minus 119906)

3= 4119901(119906 V) = 4[119901(119906

V) + 119901(V 119908)]



119901(119906119908) = 119901(119908 119906) does not hold




(119906119898 119898isinN)= [2 6] and

LIM119877minusP41(119906119898 119898isinN)

= [1 2] for sequence (119906119898



1198832


119901 (119906 119908) =

0 if 119906 ge 119908

(119906 minus 119908)2 if 119906 lt 119908

(69)

Let

119864 = [0 3) cup (3 6] (70)


2rarr [0infin) is

of the form

119869 (119906 119908) =

119901 (119906 119908) if 119864 cap 119906 119908 = 119906 119908

120583 if 119864 cap 119906 119908 = 119906 119908

(71)


36 and 119869(6 0) = 0(2)J

21 = 119869 isin J119871(119883P21)

capJ119877(119883P21)

SeeTheorem 14




119901 (119906 119908)

=

0 if 119906 ge 119908 119906 119908 cap (0 6) = 119906 119908

(119908 minus 119906)4 if 119906 lt 119908 119906 119908 cap (0 6) = 119906 119908

120574 if 119906 119908 cap (0 6) = 119906 119908

(72)


119879 (119906) =

[1 2] if 119906 isin [0 3) cup (4 6]

(4 6) if 119906 isin [3 4] (73)


Let119864 = [0 3) cup (4 6] (74)


119869 (119906 119908) =

119901 (119906 119908) if 119864 cap 119906 119908 = 119906 119908

120574 if 119864 cap 119906 119908 = 119906 119908

(75)

(1) (119883P81) where P




(2) (119883 119879) is a (119863 = D119871minusJ8112119883 = D

119877minusJ8112119883 120582 isin [2048


119909119910isin1198838 sdot

119863(119879(119909) 119879(119910)) le 120582119869(119909 119910) where

119863 (119880119882) = maxsup119906isin119880

119869 (119906119882) sup119908isin119882

119869 (119880 119908)

119880119882 isin 2119883

(76)



119864 and sup119906isin119880

inf119908isin119880

119869(119906 119908) = sup119906isin119880

119869(119906 119906) = 119901(119906 119906) =

0 = 0 Thus 4119863(119879(119909) 119879(119910)) = 0 le 120582119869(119909 119910)



inf119908isin119882

119869(119906 119908) =

sup119906isin119880

inf119908isin119882

(119908 minus 119906)4 = sup

119906isin119880(4 minus 119906)

4= 81 and

sup119908isin119882

inf119906isin119880

119869(119906 119908) = sup119908isin119882

inf119906isin119880

(119908 minus 119906)4 =

sup119908isin119882





119869(119906 119908) = sup119906isin119880

inf119908isin119882

119901(119906 119908) =

sup119908isin119882

inf119906isin119880

119869(119906 119908) = sup119908isin119882

inf119906isin119880

119901(119906 119908) = 0Therefore 8119863(119879(119909) 119879(119910)) = 0 le 120582119869(119909 119910)




exist119910isin119879(119909)

119869(119909




120573isin(0infin)119869(1199090

1199100) lt 119869(1199090 119879(1199090)) + 120573



120573isin(0infin)119869(1199090 1199100) lt 119869(1199090 119879(1199090)) + 120573


120573isin(0infin)119869(1199090 1199100) lt

119869(1199090 119879(1199090)) + 120573



119869(1199090 1199100) lt 119869(1199090 119879(1199090)) +

120573


120573isin(0infin)119869(1199090 1199100) lt 119869(1199090 119879(1199090)) + 120573


1199080isin 119883 In fact if1199080

isin 119883 and (119908119898


119898+1isin 119879(119908

119898


sup119899gt119898

119869(119908119898

119908119899

) =


sup119899gt119898

119869(119908119899

119908119898

) = 0) then forall119898ge2 119908

119898



119901(119908119908119898

) =


119898rarrinfin119901(119908

119898



119869(119908 119908119898

) =


119898rarrinfin119869(119908

119898





(119909119898 119898isinN)= (LIM119877minusP81

(119909119898 119898isinN)= ))


119898 isin N) and (119906119898

119898 isin N)


119898isin 119879(119906

119898) Then forall

119898ge2 119909119898

isin [1 2] 2 isin





following


0isin 119883




0isin 119883





119909119910isin1198838119863(119879(119909)


119901(119906119882) sup



119909119910isin1198838119863(119879(119909) 119879(119910)) le 1205820119901(119909 119910) Letting 1199090 = 2


[1 2] = 119880 119879(1199100) = (4 6) = 119882 sup119906isin[12]119901(119906 (4 6)) =

sup119906isin[12](4 minus 119906)

4= 34 = 81 and sup

119908isin(46)119901([1 2] 119908) =




119862A)


119862A whereJ



119862A)whenJ119862A = P


119862A





119860 = 1198601 cup1198602 1198601 = (0 2] 1198602 = [4 6) (77)

Let 119901 1198832


119901 (119906 119908) =

0 if 119860 cap 119906 119908 = 119906 119908

120574 if 119860 cap 119906 119908 = 119906 119908

(78)

and letJ11 = P


119879 (119906) =

1198602 for 119906 isin (0 3)

119860 for 119906 = 3

1198601 for 119906 isin (3 6)

(79)


Section 11(2) (119883 119879) is a (D



119909119910isin119883119863

119871minusP1112119883 (119879(119909) 119879(119910)) le 120582119901(119909


and maxsup119906isin119879(119909)

119901(119906 119879(119910)) sup119908isin119879(119910)



exist119910isin119879(119909)



119901 (1199090 1199100) = 119901 (1199090 119879 (1199090))

=

0 if 1199090 isin 1198601

120574 for 1199090 isin (0 3) 1198601

(80)

Therefore 119901(1199090 1199100) lt 119901(1199090 119879(1199090)) + 120573



119901 (1199090 1199100) = 119901 (1199090 119879 (1199090))

=

0 if 1199090 isin 1198602

120574 for 1199090 isin (3 6) 1198602

(81)







sup119899gt119898

119901(119908119898

119908119899


lim119898rarrinfin

119901(119908119908119898


(79)forall119898⩾1 119908

119898

isin 119879(119908119898minus1



(119908119898119898isin0cupN)




119898 isin N) sub 119879[2]

(119883) = 119860



119898 isin N) and (119906119898


119898isin 119879(119906


exist1198980isinN

forall119898⩾1198980

119909119898



and Fix(119879[2]) = 119860

(6) For (119883P11)P11 = 119901 J

11 = P11 and



isin

119883 1199081isin 119879(119908

0) and 119908

2= 119906 isin 119879(119908


and forall119898⩾3 119908

119898

= 119906 then sequence (119908119898




1198832


119901 (119906 119908) =

0 if 119906 ge 119908

(119906 minus 119908)2 if 119906 lt 119908

(82)


119879 (119906) = ([0 3) cup (3 6]) 119906 for 119906 isin [0 6] (83)

Let

119864 = [0 3) cup (3 6] (84)


2rarr [0infin) is

of the form

119869 (119906 119908) =

119901 (119906 119908) if 119864 cap 119906 119908 = 119906 119908

120583 if 119864 cap 119906 119908 = 119906 119908

(85)


119869(0 6) = 36 and 119869(6 0) = 0(2)J

21 = 119869 isin J119871(119883P21)

capJ119877(119883P21)

SeeTheorem 14




[0 1) and

119863(119880119882) = maxsup119906isin119880

119869 (119906119882) sup119908isin119882

119869 (119880 119908)

119880119882 isin 2119883

(86)




119906

cup 119882119906and

inf119908isin119882

119869 (119906 119908)

le

inf119908isin119882119906119902 (119906 119908) = 0 if 119882

119906

= 119908 isin 119882 119906 ge 119908 =

inf119908isin119882119906

(119906 minus 119908)2= 0 if 119882

119906= 119908 isin 119882 119906 lt 119908 =

(87)


cup 119880119908and

inf119906isin119880

119869 (119906 119908)

le

inf119906isin119880119908119902 (119906 119908) = 0 if 119880

119906

= 119906 isin 119880 119906 ge 119908 =

inf119906isin119880119908

(119906 minus 119908)2= 0 if 119880

119906= 119906 isin 119880 119906 lt 119908 =

(88)

By (86) 2119863(119879(119909) 119879(119910)) = 0 le 120582119869(119909 119910)

Case 2 If 119909 = 119910 = 3 then 119869(119909 119910) = 120583 and 119879(119909) =


2119863(119880119880) = 0 le 120582119869(119909 119910)



inf119908isin119882

119869(119906 119908) =


119869(119906 119908) =


inf119906isin119880


119908

cup 119880119908and

inf119906isin119880

119869 (119906 119908)

le

inf119906isin119880119908119902 (119906 119908) = 0 if 119880

119906

= 119906 isin 119880 119906 ge 119908 =

inf119906isin119880119908

(119906 minus 119908)2= 0 if 119880

119906= 119906 isin 119880 119906 lt 119908 =

(89)

Thus 2119863(119879(119909) 119879(119910)) = 0 ⩽ 120582119869(119909 119910)


(3 6]) 119910 = 119882 sub 119864 and sup119906isin119880

inf119908isin119882

119869(119906 119908) = 0 sincefor 119906 isin 119880

inf119908isin119882

119869 (119906 119908)

le

inf119908isin119882119906119902 (119906 119908) = 0 if 119882

119906

= 119908 isin 119882 119906 ge 119908 =

inf119908isin119882119906

(119906 minus 119908)2= 0 if 119882

119906= 119908 isin 119882 119906 lt 119908 =

(90)


inf119906isin119880

119869(119906 119908) = 0 since inf119906isin119880

119869(119906 119908) = 119869(119908119908) = 0 for 119908 isin 119882 Thus 2119863(119879(119909) 119879(119910)) = 0 le 120582119869(119909 119910)



exist119910isin119879(119909)

119869(119909




inf119908isin119882

119869(1199090 119908) = 0 since

inf119908isin119882

119869 (1199090 119908)

le

inf119908isin1198821199090119902 (1199090 119908) = 0 if 119882

1199090 = 119908 isin 119882 1199090 ge 119908 =

inf119908isin1198821199090

(1199090 minus 119908)2= 0 if 119882

1199090= 119908 isin 119882 1199090 lt 119908 =

(91)



12)

then 119869(1199090 1199100) lt 119869(1199090 119879(1199090)) + 120574



119869(1199090 1199100) lt 119869(1199090 119879(1199090)) + 120574




119869(1199090 119879(1199090)) + 120574


1199080


119898



sup119899gt119898

119869(119908119898

119908119899


lim119898rarrinfin

119869(119908 119908119898



1isin 119879(119908

0) = ([0 3) cup

(3 6]) 1199080 and forall

119898ge2 119908119898

isin 119879(119908119898minus1


(119908119898119898isin0cupN)

Case 2 If 1199080= 3 then 119908

1isin 119879(119908

0) = [0 3) cup (3 6] 1199082

isin

119879(1199081) = ([0 3) cup (3 6]) 119908

1 and forall

119898⩾3 119908119898

isin 119879(119908119898minus1

) sub




0isin 119883 and



119898rarrinfinsup

119899gt119898119869(119908

119898

119908119899

) =

0 of (119908119898


[2]) is a left P


119898 119898 isin N) sub 119879

[2](119883) =



119898 isin N) and (119906119898

119898 isin N)


119898isin 119879(119906


exist1198980isinN

forall119898ge1198980

119909119898


(119909119898 119898isinN)and 6 isin 119879

[2](6)

(7) ForP21 = 119901J



that 6 isin 119879[2]


0isin 119883






11 = 119901


119879 (119906) =

4 for 119906 isin (0 3)

2 for 119906 isin [3 6) (92)



119909119910isin119883119863

119871minusP111119883 (119879(119909) 119879(119910)) =

max119901(119879(119909) 119879(119910)) 119901(119879(119910) 119879(119909)) le 120582119901(119909 119910) and Fix(119879) =






119898+1= 119879(119908

119898


sup119899gt119898

119901(119908119898

119908119899

) =


119898



(119908119898119898isin0cupN)




119898 isin N) sub

119879[2]



119898 isin N) and (119906119898

119898 isin N)


119898= 119879

[2](119906

119898) then by (77) (78) and (92)



(2) isin

LIM119871minusP11(119909119898 119898isinN)

and 4 = 119879[2]





119883119860 then 119901(119906 119908) = 119901(119908 119906) = 120574 gt 0(6) For P

11 = 119901 (119883 119879) and J11 = P





119889


119889



(119883 119889) rarr (CB(119883)119867119889











References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119898 isin N) and (119908119898



119898rarrinfin

sup119899gt119898

119869120572(119906

119898 119906

119899) = 0 (8)


119898rarrinfin

sup119899gt119898

119869120572(119906

119899 119906

119898) = 0) (9)


119898rarrinfin

119869120572(119908

119898 119906

119898) = 0 (10)


119898rarrinfin

119869120572(119906

119898 119908

119898) = 0) (11)

the following holds


119898rarrinfin

119901120572(119908

119898 119906

119898) = 0 (12)


119898rarrinfin

119901120572(119906

119898 119908

119898) = 0) (13)

(B) J119871(119883P119862A)

(J119877(119883P119862A)




J119871(119883P119862A)

cap J119877(119883P119862A)


J119871(119883P119862A)



120572 120572 isin



119862A


(119883P119862A) P

119862A = and J119877(119883P119862A)

P119862A =




A where


120572ge

120575120572(119864)

2119862120572


120572(119864) = sup 119901

120572(119906 119908) 119906 119908 isin119864

(14)

If J119862A = 119869


119869120572 119883


119869120572(119906 119908) =

119901120572(119906 119908) 119894119891 119864 cap 119906 119908 = 119906 119908

120583120572

119894119891 119864 cap 119906 119908 = 119906 119908

(15)


119862A


1198691205720

(1199060 1199080) gt 1198621205720

[1198691205720

(1199060 V0) + 1198691205720

(V0 1199080)] (16)



12057201205831205720 Therefore by (14) 119901

1205720(1199060 1199080) gt

211986212057201205831205720



gt

1198621205720[119901

1205720(1199060 V0) + 120583

1205720] ge 119862

12057201205831205720


gt

1198621205720[120583

1205720+120583

1205720] = 2119862



gt

1198621205720[120583

1205720+ 119901

1205720(V0 1199080)] ge 119862

12057201205831205720


gt

1198621205720[120583

1205720+120583

1205720] = 2119862



gt

1198621205720[120583

1205720+ 120583

1205720] = 2119862



119906V119908isin119883 119869120572(119906 119908) le 119862

120572[119869

120572(119906 V) + 119869

120572(V


119898 119898 isin N) and (119908

119898



1198980=1198980(120572)isinNforall119898ge1198980

119869120572(119908

119898 119906

119898) lt 120576 (17)

Denoting 1198981015840


119898ge119898

1015840 119864 cap 119908119898 119906

119898 = 119908

119898 119906

119898



119898ge1198981015840 119901

120572(119908

119898 119906

119898) = 119869

120572(119908

119898 119906

119898) lt 120576


119898 isin N) and(119908




119898 119898 isin N)

and (119908119898



119862A isin

J119871(119883P119862A)

cap J119877(119883P119862A)

holds




119862A If P119862A


119883 that is

forall119906119908isin119883

119906 =119908


120572(119906 119908) gt 0or 119869

120572(119908 119906) gt 0

(18)

holds


120572(1199060 1199080) = 0 and 119869

120572(1199080 1199060) = 0 Then (J1) implies


120572(1199060 1199060) le 119862

120572[119869

120572(1199060 1199080)+119869

120572(1199080 1199060)] = 0 or equiv-


120572(1199060 1199060) = 119869

120572(1199080 1199060) = 0 and forall

120572isinA 119869120572(1199060

1199060) = 119869120572(1199060 1199080) = 0 Assuming that 119906

119898= 1199060 and 119908

119898= 1199080


119898rarrinfinsup

119899gt119898119869120572(119906

119898 119906

119899) =

lim119898rarrinfin

119869120572(119908

119898 119906



sup119899gt119898

119869120572(119906

119899

119906119898) = lim

119898rarrinfin119869120572(119906

119898 119908






119898rarrinfin119901120572(119908

119898 119906

119898) = lim

119898rarrinfin119901120572(119906

119898 119908

119898) = 0 or


120572(1199080 1199060) = 119901

120572(1199060 1199080) =








119898 119898 isin N) sub 119883 is left


119898rarrinfinsup

119899gt119898119869120572(119906

119898

119906119899) = 0 (forall


sup119899gt119898

119869120572(119906

119899 119906

119898) = 0)

(B) Let 119906 isin 119883 and let (119906119898





(119906119898 119898isinN)= ) where


= 119909


119898rarrinfin

119869120572(119909 119906

119898) = 0



119898rarrinfin

119869120572(119906

119898 119909) = 0)

(19)


119898 isin N) sub


(119906119898 119898isinN)=







= )) then (119883P119862A) is











sub



119898 isin N) of (119906119898

119898 isin N)



119898



) is a singleton







(A) Given 1199080



119898


119898isin0cupN 119908119898+1

isin 119879(119908119898





119898rarrinfin

sup119899gt119898

119869120572(119908

119898

119908119899

) = 0


119898rarrinfin

sup119899gt119898

119869120572(119908

119899

119908119898

) = 0)

(20)



isin

119883




119862A)



holds




[119896]



119898 119898 isin N) in 119879

[119896]





119898 119898 isin N) and (119906


forall119898isinN V

119898isin 119879

[119896]





119879[119896]

(119909) (119909 isin 119879[119896]

(119909))







119862A Let120582 = 120582




119909isin119883forall119881isin2119883 119869

120572(119909 119881) = inf 119869

120572(119909 V) V isin119881

and 119869120572(119881 119909) = inf 119869

120572(V 119909) V isin119881

(21)


(A) LetJ119862A isin J119871

(119883P119862A) If


119880119882isin2119883 119863119871minusJ119862A

12119883120572 (119880119882)


119869120572(119906119882) sup

119908isin119882

119869120572(119880 119908)


119880119882isin2119883 119863119871minusJ119862A

22119883120572 (119880119882)


119869120572(119906119882) sup

119908isin119882

119869120572(119908 119880)


119880119882isin2119883 119863119871minusJ119862A

32119883120572 (119880119882) = sup119906isin119880

119869120572(119906119882)

(22)


1205782119883 = 119863119871minusJ119862A





119909119910isin119883119862

120572sdot 119863

119871minusJ119862A

1205782119883120572 (119879 (119909) 119879 (119910))

le 120582120572119869120572(119909 119910)

(23)


1205782119883 120582)-quasi-119888119900119899119905119903119886119888119905119894119900119899on 119883

(B) LetJ119862A isin J119877

(119883P119862A) If


119880119882isin2119883 119863119877minusJ119862A

12119883120572 (119880119882)


119869120572(119906119882) sup

119908isin119882

119869120572(119880 119908)


119880119882isin2119883 119863119877minusJ119862A

22119883120572 (119880119882)


119869120572(119906119882) sup

119908isin119882

119869120572(119908 119880)


119880119882isin2119883 119863119877minusJ119862A

32119883120572 (119880119882) = sup119906isin119880

119869120572(119906119882)

(24)


1205782119883 = 119863119877minusJ119862A





119909119910isin119883119862

120572sdot 119863

119877minusJ119862A

1205782119883120572 (119879 (119909) 119879 (119910))

le 120582120572119869120572(119909 119910)

(25)




1205782119883 and D119877minusJ119862A




1205782119883



119877minusJ119862A






120572120572isinA isin [0 1)A




119871minusJ119862A


(D119877minusJ119862A





A



120572(119909 119910) lt 119869

120572(119909 119879 (119909)) + 120573

120572 (26)


120572(119910 119909) lt 119869

120572(119879 (119909) 119909) + 120573

120572) (27)


119898



119898+1isin 119879(119908

119898




[119896]

) is left (right)P




119898isin0cupN 119908119898+1

isin 119879(119908119898


) such that (119908119898






0isin 119883 and (119883 119879

[119896]




120572 119883


choose

119903 = 119903120572120572isinA

isin (0infin)A (28)


such that


120572(119908

0 119879 (119908

0)) lt(1minus

120582120572

119862120572

) 119903120572 (29)

Put


(0)120572

=(1minus120582120572

119862120572

) 119903120572minus 119869

120572(119908

0 119879 (119908

0)) (30)


= 120573(0)120572

120572isinA isin


1isin 119879(119908

0) such that


120572(119908

0 119908

1) lt 119869

120572(119908

0 119879 (119908

0)) + 120573

(0)120572

(31)



120572(119908

0 119908

1) lt(1minus

120582120572

119862120572

) 119903120572 (32)

Put now


(1)120572

=(120582120572

119862120572

)[(1minus120582120572

119862120572

) 119903120572minus 119869

120572(119908

0 119908

1)] (33)


(1)120572



2isin 119879(119908

1) such that


120572(119908

1 119908

2) lt 119869

120572(119908

1 119879 (119908

1)) + 120573

(1)120572

(34)

Observe that


120572(119908

1 119908

2) lt(

120582120572

119862120572

)(1minus120582120572

119862120572

) 119903120572 (35)



120572(119908

1 119908

2) lt 119869

120572(119908

1 119879 (119908

1)) + 120573

(1)120572

le sup 119869120572(119906 119879 (119908

1)) 119906 isin119879 (119908

0) + 120573

(1)120572

le119863119871minusJ119862A

1205782119883120572 (119879 (1199080) 119879 (119908

1)) + 120573

(1)120572

le(120582120572

119862120572

) 119869120572(119908

0 119908

1) + 120573

(1)120572

=(120582120572

119862120572

)(1minus120582120572

119862120572

) 119903120572

(36)



(2)120572

=(120582120572

119862120572

)[(120582120572

119862120572

)(1minus120582120572

119862120572

) 119903120572minus 119869

120572(119908

1 119908

2)]

(37)

Then in view of (35) 120573(2)= 120573

(2)120572



3isin

119879(1199082) such that


120572(119908

2 119908

3) lt 119869

120572(119908

2 119879 (119908

2)) + 120573

(2)120572

(38)



120572(119908

2 119908

3) lt(

120582120572

119862120572

)

2

(1minus120582120572

119862120572

) 119903120572 (39)



120572(119908

2 119908

3) lt 119869

120572(119908

2 119879 (119908

2)) + 120573

(2)120572

⩽ sup119906isin119879(119908

1)

119869120572(119906 119879 (119908

2)) + 120573

(2)120572

le119863119871minusJ119862A

1205782119883120572 (119879 (1199081) 119879 (119908

2)) + 120573

(2)120572

le(120582120572

119862120572

) 119869120572(119908

1 119908

2) + 120573

(2)120572

=(120582120572

119862120572

)

2

(1minus120582120572

119862120572

) 119903120572

(40)





119898+1isin119879 (119908

119898

) (41)


(119898)

=

120573(119898)

120572120572isinA

where


119898isinN 120573(119898)

120572=(

120582120572

119862120572

)

sdot [(120582120572

119862120572

)

119898minus1(1minus

120582120572

119862120572

) 119903120572minus 119869

120572(119908

119898minus1 119908

119898

)]

(42)


(119898)

isin (0infin)A


119898isinN 119869120572(119908

119898

119908119898+1

) lt 119869120572(119908

119898

119879 (119908119898

))

+ 120573(119898)

120572

(43)


119898isinN 119869120572(119908

119898

119908119898+1

)

lt(120582120572

119862120572

)

119898

(1minus120582120572

119862120572

) 119903120572

(44)



forall120572isinA

119869120572(119908

119898

119908119899

) ⩽119862120572119869120572(119908

119898

119908119898+1

)

+1198622120572119869120572(119908

119898+1 119908


+ 119862119899minus119898minus1120572

119869120572(119908

119899minus2 119908

119899minus1) +119862

119899minus119898minus1120572

119869120572(119908

119899minus1 119908

119899

)

=


119895=0119862119895+1120572

119869120572(119908

119898+119895

119908119898+119895+1

)

+119862119899minus119898minus1120572

119869120572(119908

119899minus1 119908

119899

)

(45)


119869120572(119908

119898

119908119899

) lt (1minus120582120572

119862120572

)

sdot 119903120572

[

[


119895=0119862119895+1120572

(120582120572

119862120572

)

119898+119895

+119862119899minus119898minus1120572

(120582120572

119862120572

)

119899minus2]

]

= (1minus120582120572

119862120572

)

sdot 119903120572

[

[

119862120572(

120582120572

119862120572

)

119898 119899minus119898minus2sum

119895=0120582119895

120572+(

119862120572

1205822120572

)120582119899

120572

119862119898

120572

]

]

(46)



119898isin0cupN 119908119898+1

isin119879 (119908119898

) (47)


120572le 120582

119898

120572


119898rarrinfin

sup119899gt119898

119869120572(119908

119898

119908119899


sup119899gt119898

(1minus120582120572

119862120572

)

sdot 119903120572[119862

120572(

120582120572

119862120572

)

119898

(1 minus 120582120572)minus1

+(119862120572

1205822120572

)120582119899

120572

119862119898

120572

]


(1minus120582120572

119862120572

) 119903120572[119862

120572(

120582120572

119862120572

)

119898

(1minus120582120572)minus1

+(119862120572

1205822120572

)(120582120572

119862120572

)

119898

]= 0

(48)


0isin 119883 by



119898rarrinfin

119869120572(119908 119908

119898

) = 0 (49)


= 119908119898 and 119908

119898= 119908 for 119898 isin N by



119898 isin N) and (119908119898



119898rarrinfin

119901120572(119908 119908

119898


119901120572(119908

119898 119906

119898) = 0 (50)



[119896]



119908(119898+1)119896

isin 119879[119896]

(119908119898119896


=

119908119898minus1+119896

119898 isin N) we see that (119909119898

119898 isin N) sub 119879[119896]

(119883)LIM119871minusP119862A


119898=

119908(119898+1)119896

119898 isin N) sub 119879[119896]

(119883) and (119906119898

= 119908119898119896

119898 isin N) sub

119879[119896]


119898isin 119879

[119896]


of (119909119898








[119896] is left P119862A-



such that 119908 isin 119879[119896]

(119908)



119862A)









119862A isin J119871(119883P119862A)


-


= 119863119871minusJ119862A120578119883120572

1198832



119906119908isin119883119863

119871minusJ119862A1119883120572 (119906 119908)

=max 119869120572(119906 119908) 119869

120572(119908 119906)


119906119908isin119883119863

119871minusJ119862A2119883120572 (119906 119908) = 119869

120572(119906 119908)

(51)





119909119910isin119883119862

120572sdot 119863

119871minusJ119862A120578119883120572

(119879 (119909) 119879 (119910))

le 120582120572119869120572(119909 119910)

(52)

(B) If J119862A isin J119877


D119877minusJ119862A119883120578


= 119863119877minusJ119862A120578119883120572

1198832

rarr



119906119908isin119883119863

119877minusJ119862A1119883120572 (119906 119908)

=max 119869120572(119906 119908) 119869

120572(119908 119906)


119906119908isin119883119863

119877minusJ119862A2119883120572 (119906 119908) = 119869

120572(119906 119908)

(53)




119909119910isin119883119862

120572sdot 119863

119877minusJ119862A120578119883120572

(119879 (119909) 119879 (119910))

le 120582120572119869120572(119909 119910)

(54)






(A) Given 1199080



119898

= 119879[119898]

(1199080)


(119908119898119898isin0cupN) = )

whenever


119898rarrinfin

sup119899gt119898

119869120572(119908

119898

119908119899

) = 0


119898rarrinfin

sup119899gt119898

119869120572(119908

119899

119908119898

) = 0)

(55)



isin 119883









[119896]

) is


(119909119898

119898 isin N) in 119879[119896]





119898 119898 isin N) and (119906


forall119898isinN V

119898= 119879

[119896]





119879[119896]

(119909) (119909 = 119879[119896]

(119909))


119862A)



120572120572isinA isin [0 1)A




119871minusJ119862A119883120578


119877minusJ119862A119883120578




(119908119898

= 119879[119898]



[119896]

) is left(right)P




119898

= 119879[119898]




120572(V 119879 (V)) = 119869

120572(119879 (V) V) = 0 (56)




) is left (right)P


Fix (119879[119896]

) = Fix (119879) = 119908 (57)


= 119879[119898]




120572(119908 119908) = 0 (58)




[119896]

(V) = 119879[2119896]

(V) 119879(V) =

119879[2119896]


0 lt 1198691205720

(V 119879 (V)) = 1198691205720

(119879[2119896]

(V) 119879[2119896](119879 (V)))

le 119863119871minusJ119862A1205781198831205720

(119879[2119896]

(V) 119879[2119896](119879 (V)))


le (1205821205720

1198621205720

)1198691205720

(119879[2119896minus1]

(V) 119879[2119896minus1](119879 (V)))

le (1205821205720

1198621205720

) sdot119863119871minusJ119862A1205781198831205720

(119879[2119896minus1]

(V) 119879[2119896minus1](119879 (V)))

le (1205821205720

1198621205720

)

2

1198691205720

(119879[2119896minus2]


le (1205821205720

1198621205720

)

2119896

1198691205720

(V 119879 (V)) lt 1198691205720

(V 119879 (V))

(59)



120572(V 119879 (V)) = 0 (60)




119879[119896]

(V) = 119879[2119896]


0 lt 1198691205720

(119879 (V) V) = 1198691205720

(119879[119896+1]

(V) 119879[2119896](V))

le

119896minus2sum

119898=1119862119898

12057201198691205720

(119879[119896+119898]

(V) 119879[119896+119898+1](V))

+119862119896minus21205720

1198691205720

(119879[2119896minus1]

(V) 119879[2119896](V))

le

119896minus2sum

119898=1119862119898

1205720sdot 119863

119871minusJ119862A1205781198831205720

(119879[119896+119898]

(V) 119879[119896+119898+1](V))

+119862119896minus21205720

sdot 119863119871minusJ119862A1205781198831205720

(119879[2119896minus1]

(V) 119879[2119896](V))

le

119896minus2sum

119898=1119862119898

1205720(

1205821205720

1198621205720

)

119896+119898

1198691205720

(V 119879 (V))

+119862119896minus21205720

(1205821205720

1198621205720

)

2119896minus1

1198691205720

(V 119879 (V)) = 0

(61)



120572(119879 (V) V) = 0 (62)






)


120572 120572 isin A is


120572(119879(V) V) gt



120572(119879(V) V) gt 0 or 119869

120572(V 119879(V)) gt 0 However by


120572(V V) = 0 In fact


120572(V V) le 119862

120572[119869

120572(V 119879(V))+119869

120572(119879(V) V)] = 0




1199011205720(119906 119908) gt 0 or 119901

1205720(119908 119906) gt 0 By applying


1198691205720(119906 119908) gt 0 or 119869

1205720(119908 119906) gt


exist1205720isinA

[1198691205720

(119906 119908) gt 0 1198691205720

(119906 119908)

= 1198691205720

(119879 (119906) 119879 (119908)) le119863119871minusJ119862A1205781198831205720

(119879 (119906) 119879 (119908))

le(1205821205720

1198621205720

)1198691205720

(119906 119908) lt 1198691205720

(119906 119908)] or [1198691205720

(119908 119906)

gt 0 1198691205720

(119908 119906) = 1198691205720

(119879 (119908) 119879 (119906))

le119863119871minusJ119862A1205781198831205720

(119879 (119908) 119879 (119906)) le(1205821205720

1198621205720

)1198691205720

(119908 119906)

lt 1198691205720

(119908 119906)]

(64)




rarr [0infin)

be of the form

119901 (119906 119908)

=

0 if 119906 ge 119908 119906 119908 cap (0 6) = 119906 119908

(119908 minus 119906)4 if 119906 lt 119908 119906 119908 cap (0 6) = 119906 119908

120574 if 119906 119908 cap (0 6) = 119906 119908

(65)


space In fact





8(119908 minus V)4 = 8[119901(119906 V) + 119901(V 119908)]


(119908 minus 119906)4 and 119891(V0) = min



and V0 = (119906 + 119908)2


119906119908isin(06)119906lt119908(119908 minus 119906)4

=


119906leVle1199088[119901(119906 V) + 119901(V 119908)] =

sup119906119908isin(06)119906lt119908min


(6 minus 3)4] = 8[81 + 81] = 1296




4lt

8(V minus 119906)4= 8[119901(119906 V) + 119901(V 119908)]


1296 le 8[119901(119906 V) + 119901(V 119908)] = 8[120574 + 120574]


8120574 le 8[119901(119906 V) + 119901(V 119908)]



119901(119906119908) = 119901(119908 119906) does not hold




119898= 3





(119906119898 119898isinN)= [0 3]



119901 (119906 119908) =

0 if 119860 cap 119906 119908 = 119906 119908

120574 if 119860 cap 119906 119908 = 119906 119908

(67)









119901(119906 119906) = 0 does not hold(3)P


(119906119898 119898isinN)= LIM119877minusP11

(119906119898 119898isinN)= 119860 for





119901 (119906 119908) =

0 if 119906 ge 119908

(119908 minus 119906)3 if 119906 lt 119908

(68)






4119901(V 119908) = 4[119901(119906 V) + 119901(V 119908)]


119891(V0) = min119906⩽V⩽119908119891(V) = (119908 minus 119906)

3 where V0 = (119906 + 119908)2 is


4(119908 minus 119906)[1199082+ 119908119906 + 119906

2+ 3V2 minus 3V(119908 + 119906)]



3lt 4(V minus 119906)

3= 4119901(119906 V) = 4[119901(119906

V) + 119901(V 119908)]



119901(119906119908) = 119901(119908 119906) does not hold




(119906119898 119898isinN)= [2 6] and

LIM119877minusP41(119906119898 119898isinN)

= [1 2] for sequence (119906119898



1198832


119901 (119906 119908) =

0 if 119906 ge 119908

(119906 minus 119908)2 if 119906 lt 119908

(69)

Let

119864 = [0 3) cup (3 6] (70)


2rarr [0infin) is

of the form

119869 (119906 119908) =

119901 (119906 119908) if 119864 cap 119906 119908 = 119906 119908

120583 if 119864 cap 119906 119908 = 119906 119908

(71)


36 and 119869(6 0) = 0(2)J

21 = 119869 isin J119871(119883P21)

capJ119877(119883P21)

SeeTheorem 14




119901 (119906 119908)

=

0 if 119906 ge 119908 119906 119908 cap (0 6) = 119906 119908

(119908 minus 119906)4 if 119906 lt 119908 119906 119908 cap (0 6) = 119906 119908

120574 if 119906 119908 cap (0 6) = 119906 119908

(72)


119879 (119906) =

[1 2] if 119906 isin [0 3) cup (4 6]

(4 6) if 119906 isin [3 4] (73)


Let119864 = [0 3) cup (4 6] (74)


119869 (119906 119908) =

119901 (119906 119908) if 119864 cap 119906 119908 = 119906 119908

120574 if 119864 cap 119906 119908 = 119906 119908

(75)

(1) (119883P81) where P




(2) (119883 119879) is a (119863 = D119871minusJ8112119883 = D

119877minusJ8112119883 120582 isin [2048


119909119910isin1198838 sdot

119863(119879(119909) 119879(119910)) le 120582119869(119909 119910) where

119863 (119880119882) = maxsup119906isin119880

119869 (119906119882) sup119908isin119882

119869 (119880 119908)

119880119882 isin 2119883

(76)



119864 and sup119906isin119880

inf119908isin119880

119869(119906 119908) = sup119906isin119880

119869(119906 119906) = 119901(119906 119906) =

0 = 0 Thus 4119863(119879(119909) 119879(119910)) = 0 le 120582119869(119909 119910)



inf119908isin119882

119869(119906 119908) =

sup119906isin119880

inf119908isin119882

(119908 minus 119906)4 = sup

119906isin119880(4 minus 119906)

4= 81 and

sup119908isin119882

inf119906isin119880

119869(119906 119908) = sup119908isin119882

inf119906isin119880

(119908 minus 119906)4 =

sup119908isin119882





119869(119906 119908) = sup119906isin119880

inf119908isin119882

119901(119906 119908) =

sup119908isin119882

inf119906isin119880

119869(119906 119908) = sup119908isin119882

inf119906isin119880

119901(119906 119908) = 0Therefore 8119863(119879(119909) 119879(119910)) = 0 le 120582119869(119909 119910)




exist119910isin119879(119909)

119869(119909




120573isin(0infin)119869(1199090

1199100) lt 119869(1199090 119879(1199090)) + 120573



120573isin(0infin)119869(1199090 1199100) lt 119869(1199090 119879(1199090)) + 120573


120573isin(0infin)119869(1199090 1199100) lt

119869(1199090 119879(1199090)) + 120573



119869(1199090 1199100) lt 119869(1199090 119879(1199090)) +

120573


120573isin(0infin)119869(1199090 1199100) lt 119869(1199090 119879(1199090)) + 120573


1199080isin 119883 In fact if1199080

isin 119883 and (119908119898


119898+1isin 119879(119908

119898


sup119899gt119898

119869(119908119898

119908119899

) =


sup119899gt119898

119869(119908119899

119908119898

) = 0) then forall119898ge2 119908

119898



119901(119908119908119898

) =


119898rarrinfin119901(119908

119898



119869(119908 119908119898

) =


119898rarrinfin119869(119908

119898





(119909119898 119898isinN)= (LIM119877minusP81

(119909119898 119898isinN)= ))


119898 isin N) and (119906119898

119898 isin N)


119898isin 119879(119906

119898) Then forall

119898ge2 119909119898

isin [1 2] 2 isin





following


0isin 119883




0isin 119883





119909119910isin1198838119863(119879(119909)


119901(119906119882) sup



119909119910isin1198838119863(119879(119909) 119879(119910)) le 1205820119901(119909 119910) Letting 1199090 = 2


[1 2] = 119880 119879(1199100) = (4 6) = 119882 sup119906isin[12]119901(119906 (4 6)) =

sup119906isin[12](4 minus 119906)

4= 34 = 81 and sup

119908isin(46)119901([1 2] 119908) =




119862A)


119862A whereJ



119862A)whenJ119862A = P


119862A





119860 = 1198601 cup1198602 1198601 = (0 2] 1198602 = [4 6) (77)

Let 119901 1198832


119901 (119906 119908) =

0 if 119860 cap 119906 119908 = 119906 119908

120574 if 119860 cap 119906 119908 = 119906 119908

(78)

and letJ11 = P


119879 (119906) =

1198602 for 119906 isin (0 3)

119860 for 119906 = 3

1198601 for 119906 isin (3 6)

(79)


Section 11(2) (119883 119879) is a (D



119909119910isin119883119863

119871minusP1112119883 (119879(119909) 119879(119910)) le 120582119901(119909


and maxsup119906isin119879(119909)

119901(119906 119879(119910)) sup119908isin119879(119910)



exist119910isin119879(119909)



119901 (1199090 1199100) = 119901 (1199090 119879 (1199090))

=

0 if 1199090 isin 1198601

120574 for 1199090 isin (0 3) 1198601

(80)

Therefore 119901(1199090 1199100) lt 119901(1199090 119879(1199090)) + 120573



119901 (1199090 1199100) = 119901 (1199090 119879 (1199090))

=

0 if 1199090 isin 1198602

120574 for 1199090 isin (3 6) 1198602

(81)







sup119899gt119898

119901(119908119898

119908119899


lim119898rarrinfin

119901(119908119908119898


(79)forall119898⩾1 119908

119898

isin 119879(119908119898minus1



(119908119898119898isin0cupN)




119898 isin N) sub 119879[2]

(119883) = 119860



119898 isin N) and (119906119898


119898isin 119879(119906


exist1198980isinN

forall119898⩾1198980

119909119898



and Fix(119879[2]) = 119860

(6) For (119883P11)P11 = 119901 J

11 = P11 and



isin

119883 1199081isin 119879(119908

0) and 119908

2= 119906 isin 119879(119908


and forall119898⩾3 119908

119898

= 119906 then sequence (119908119898




1198832


119901 (119906 119908) =

0 if 119906 ge 119908

(119906 minus 119908)2 if 119906 lt 119908

(82)


119879 (119906) = ([0 3) cup (3 6]) 119906 for 119906 isin [0 6] (83)

Let

119864 = [0 3) cup (3 6] (84)


2rarr [0infin) is

of the form

119869 (119906 119908) =

119901 (119906 119908) if 119864 cap 119906 119908 = 119906 119908

120583 if 119864 cap 119906 119908 = 119906 119908

(85)


119869(0 6) = 36 and 119869(6 0) = 0(2)J

21 = 119869 isin J119871(119883P21)

capJ119877(119883P21)

SeeTheorem 14




[0 1) and

119863(119880119882) = maxsup119906isin119880

119869 (119906119882) sup119908isin119882

119869 (119880 119908)

119880119882 isin 2119883

(86)




119906

cup 119882119906and

inf119908isin119882

119869 (119906 119908)

le

inf119908isin119882119906119902 (119906 119908) = 0 if 119882

119906

= 119908 isin 119882 119906 ge 119908 =

inf119908isin119882119906

(119906 minus 119908)2= 0 if 119882

119906= 119908 isin 119882 119906 lt 119908 =

(87)


cup 119880119908and

inf119906isin119880

119869 (119906 119908)

le

inf119906isin119880119908119902 (119906 119908) = 0 if 119880

119906

= 119906 isin 119880 119906 ge 119908 =

inf119906isin119880119908

(119906 minus 119908)2= 0 if 119880

119906= 119906 isin 119880 119906 lt 119908 =

(88)

By (86) 2119863(119879(119909) 119879(119910)) = 0 le 120582119869(119909 119910)

Case 2 If 119909 = 119910 = 3 then 119869(119909 119910) = 120583 and 119879(119909) =


2119863(119880119880) = 0 le 120582119869(119909 119910)



inf119908isin119882

119869(119906 119908) =


119869(119906 119908) =


inf119906isin119880


119908

cup 119880119908and

inf119906isin119880

119869 (119906 119908)

le

inf119906isin119880119908119902 (119906 119908) = 0 if 119880

119906

= 119906 isin 119880 119906 ge 119908 =

inf119906isin119880119908

(119906 minus 119908)2= 0 if 119880

119906= 119906 isin 119880 119906 lt 119908 =

(89)

Thus 2119863(119879(119909) 119879(119910)) = 0 ⩽ 120582119869(119909 119910)


(3 6]) 119910 = 119882 sub 119864 and sup119906isin119880

inf119908isin119882

119869(119906 119908) = 0 sincefor 119906 isin 119880

inf119908isin119882

119869 (119906 119908)

le

inf119908isin119882119906119902 (119906 119908) = 0 if 119882

119906

= 119908 isin 119882 119906 ge 119908 =

inf119908isin119882119906

(119906 minus 119908)2= 0 if 119882

119906= 119908 isin 119882 119906 lt 119908 =

(90)


inf119906isin119880

119869(119906 119908) = 0 since inf119906isin119880

119869(119906 119908) = 119869(119908119908) = 0 for 119908 isin 119882 Thus 2119863(119879(119909) 119879(119910)) = 0 le 120582119869(119909 119910)



exist119910isin119879(119909)

119869(119909




inf119908isin119882

119869(1199090 119908) = 0 since

inf119908isin119882

119869 (1199090 119908)

le

inf119908isin1198821199090119902 (1199090 119908) = 0 if 119882

1199090 = 119908 isin 119882 1199090 ge 119908 =

inf119908isin1198821199090

(1199090 minus 119908)2= 0 if 119882

1199090= 119908 isin 119882 1199090 lt 119908 =

(91)



12)

then 119869(1199090 1199100) lt 119869(1199090 119879(1199090)) + 120574



119869(1199090 1199100) lt 119869(1199090 119879(1199090)) + 120574




119869(1199090 119879(1199090)) + 120574


1199080


119898



sup119899gt119898

119869(119908119898

119908119899


lim119898rarrinfin

119869(119908 119908119898



1isin 119879(119908

0) = ([0 3) cup

(3 6]) 1199080 and forall

119898ge2 119908119898

isin 119879(119908119898minus1


(119908119898119898isin0cupN)

Case 2 If 1199080= 3 then 119908

1isin 119879(119908

0) = [0 3) cup (3 6] 1199082

isin

119879(1199081) = ([0 3) cup (3 6]) 119908

1 and forall

119898⩾3 119908119898

isin 119879(119908119898minus1

) sub




0isin 119883 and



119898rarrinfinsup

119899gt119898119869(119908

119898

119908119899

) =

0 of (119908119898


[2]) is a left P


119898 119898 isin N) sub 119879

[2](119883) =



119898 isin N) and (119906119898

119898 isin N)


119898isin 119879(119906


exist1198980isinN

forall119898ge1198980

119909119898


(119909119898 119898isinN)and 6 isin 119879

[2](6)

(7) ForP21 = 119901J



that 6 isin 119879[2]


0isin 119883






11 = 119901


119879 (119906) =

4 for 119906 isin (0 3)

2 for 119906 isin [3 6) (92)



119909119910isin119883119863

119871minusP111119883 (119879(119909) 119879(119910)) =

max119901(119879(119909) 119879(119910)) 119901(119879(119910) 119879(119909)) le 120582119901(119909 119910) and Fix(119879) =






119898+1= 119879(119908

119898


sup119899gt119898

119901(119908119898

119908119899

) =


119898



(119908119898119898isin0cupN)




119898 isin N) sub

119879[2]



119898 isin N) and (119906119898

119898 isin N)


119898= 119879

[2](119906

119898) then by (77) (78) and (92)



(2) isin

LIM119871minusP11(119909119898 119898isinN)

and 4 = 119879[2]





119883119860 then 119901(119906 119908) = 119901(119908 119906) = 120574 gt 0(6) For P

11 = 119901 (119883 119879) and J11 = P





119889


119889



(119883 119889) rarr (CB(119883)119867119889











References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119898rarrinfin119901120572(119908

119898 119906

119898) = lim

119898rarrinfin119901120572(119906

119898 119908

119898) = 0 or


120572(1199080 1199060) = 119901

120572(1199060 1199080) =








119898 119898 isin N) sub 119883 is left


119898rarrinfinsup

119899gt119898119869120572(119906

119898

119906119899) = 0 (forall


sup119899gt119898

119869120572(119906

119899 119906

119898) = 0)

(B) Let 119906 isin 119883 and let (119906119898





(119906119898 119898isinN)= ) where


= 119909


119898rarrinfin

119869120572(119909 119906

119898) = 0



119898rarrinfin

119869120572(119906

119898 119909) = 0)

(19)


119898 isin N) sub


(119906119898 119898isinN)=







= )) then (119883P119862A) is











sub



119898 isin N) of (119906119898

119898 isin N)



119898



) is a singleton







(A) Given 1199080



119898


119898isin0cupN 119908119898+1

isin 119879(119908119898





119898rarrinfin

sup119899gt119898

119869120572(119908

119898

119908119899

) = 0


119898rarrinfin

sup119899gt119898

119869120572(119908

119899

119908119898

) = 0)

(20)



isin

119883




119862A)



holds




[119896]



119898 119898 isin N) in 119879

[119896]





119898 119898 isin N) and (119906


forall119898isinN V

119898isin 119879

[119896]





119879[119896]

(119909) (119909 isin 119879[119896]

(119909))







119862A Let120582 = 120582




119909isin119883forall119881isin2119883 119869

120572(119909 119881) = inf 119869

120572(119909 V) V isin119881

and 119869120572(119881 119909) = inf 119869

120572(V 119909) V isin119881

(21)


(A) LetJ119862A isin J119871

(119883P119862A) If


119880119882isin2119883 119863119871minusJ119862A

12119883120572 (119880119882)


119869120572(119906119882) sup

119908isin119882

119869120572(119880 119908)


119880119882isin2119883 119863119871minusJ119862A

22119883120572 (119880119882)


119869120572(119906119882) sup

119908isin119882

119869120572(119908 119880)


119880119882isin2119883 119863119871minusJ119862A

32119883120572 (119880119882) = sup119906isin119880

119869120572(119906119882)

(22)


1205782119883 = 119863119871minusJ119862A





119909119910isin119883119862

120572sdot 119863

119871minusJ119862A

1205782119883120572 (119879 (119909) 119879 (119910))

le 120582120572119869120572(119909 119910)

(23)


1205782119883 120582)-quasi-119888119900119899119905119903119886119888119905119894119900119899on 119883

(B) LetJ119862A isin J119877

(119883P119862A) If


119880119882isin2119883 119863119877minusJ119862A

12119883120572 (119880119882)


119869120572(119906119882) sup

119908isin119882

119869120572(119880 119908)


119880119882isin2119883 119863119877minusJ119862A

22119883120572 (119880119882)


119869120572(119906119882) sup

119908isin119882

119869120572(119908 119880)


119880119882isin2119883 119863119877minusJ119862A

32119883120572 (119880119882) = sup119906isin119880

119869120572(119906119882)

(24)


1205782119883 = 119863119877minusJ119862A





119909119910isin119883119862

120572sdot 119863

119877minusJ119862A

1205782119883120572 (119879 (119909) 119879 (119910))

le 120582120572119869120572(119909 119910)

(25)




1205782119883 and D119877minusJ119862A




1205782119883



119877minusJ119862A






120572120572isinA isin [0 1)A




119871minusJ119862A


(D119877minusJ119862A





A



120572(119909 119910) lt 119869

120572(119909 119879 (119909)) + 120573

120572 (26)


120572(119910 119909) lt 119869

120572(119879 (119909) 119909) + 120573

120572) (27)


119898



119898+1isin 119879(119908

119898




[119896]

) is left (right)P




119898isin0cupN 119908119898+1

isin 119879(119908119898


) such that (119908119898






0isin 119883 and (119883 119879

[119896]




120572 119883


choose

119903 = 119903120572120572isinA

isin (0infin)A (28)


such that


120572(119908

0 119879 (119908

0)) lt(1minus

120582120572

119862120572

) 119903120572 (29)

Put


(0)120572

=(1minus120582120572

119862120572

) 119903120572minus 119869

120572(119908

0 119879 (119908

0)) (30)


= 120573(0)120572

120572isinA isin


1isin 119879(119908

0) such that


120572(119908

0 119908

1) lt 119869

120572(119908

0 119879 (119908

0)) + 120573

(0)120572

(31)



120572(119908

0 119908

1) lt(1minus

120582120572

119862120572

) 119903120572 (32)

Put now


(1)120572

=(120582120572

119862120572

)[(1minus120582120572

119862120572

) 119903120572minus 119869

120572(119908

0 119908

1)] (33)


(1)120572



2isin 119879(119908

1) such that


120572(119908

1 119908

2) lt 119869

120572(119908

1 119879 (119908

1)) + 120573

(1)120572

(34)

Observe that


120572(119908

1 119908

2) lt(

120582120572

119862120572

)(1minus120582120572

119862120572

) 119903120572 (35)



120572(119908

1 119908

2) lt 119869

120572(119908

1 119879 (119908

1)) + 120573

(1)120572

le sup 119869120572(119906 119879 (119908

1)) 119906 isin119879 (119908

0) + 120573

(1)120572

le119863119871minusJ119862A

1205782119883120572 (119879 (1199080) 119879 (119908

1)) + 120573

(1)120572

le(120582120572

119862120572

) 119869120572(119908

0 119908

1) + 120573

(1)120572

=(120582120572

119862120572

)(1minus120582120572

119862120572

) 119903120572

(36)



(2)120572

=(120582120572

119862120572

)[(120582120572

119862120572

)(1minus120582120572

119862120572

) 119903120572minus 119869

120572(119908

1 119908

2)]

(37)

Then in view of (35) 120573(2)= 120573

(2)120572



3isin

119879(1199082) such that


120572(119908

2 119908

3) lt 119869

120572(119908

2 119879 (119908

2)) + 120573

(2)120572

(38)



120572(119908

2 119908

3) lt(

120582120572

119862120572

)

2

(1minus120582120572

119862120572

) 119903120572 (39)



120572(119908

2 119908

3) lt 119869

120572(119908

2 119879 (119908

2)) + 120573

(2)120572

⩽ sup119906isin119879(119908

1)

119869120572(119906 119879 (119908

2)) + 120573

(2)120572

le119863119871minusJ119862A

1205782119883120572 (119879 (1199081) 119879 (119908

2)) + 120573

(2)120572

le(120582120572

119862120572

) 119869120572(119908

1 119908

2) + 120573

(2)120572

=(120582120572

119862120572

)

2

(1minus120582120572

119862120572

) 119903120572

(40)





119898+1isin119879 (119908

119898

) (41)


(119898)

=

120573(119898)

120572120572isinA

where


119898isinN 120573(119898)

120572=(

120582120572

119862120572

)

sdot [(120582120572

119862120572

)

119898minus1(1minus

120582120572

119862120572

) 119903120572minus 119869

120572(119908

119898minus1 119908

119898

)]

(42)


(119898)

isin (0infin)A


119898isinN 119869120572(119908

119898

119908119898+1

) lt 119869120572(119908

119898

119879 (119908119898

))

+ 120573(119898)

120572

(43)


119898isinN 119869120572(119908

119898

119908119898+1

)

lt(120582120572

119862120572

)

119898

(1minus120582120572

119862120572

) 119903120572

(44)



forall120572isinA

119869120572(119908

119898

119908119899

) ⩽119862120572119869120572(119908

119898

119908119898+1

)

+1198622120572119869120572(119908

119898+1 119908


+ 119862119899minus119898minus1120572

119869120572(119908

119899minus2 119908

119899minus1) +119862

119899minus119898minus1120572

119869120572(119908

119899minus1 119908

119899

)

=


119895=0119862119895+1120572

119869120572(119908

119898+119895

119908119898+119895+1

)

+119862119899minus119898minus1120572

119869120572(119908

119899minus1 119908

119899

)

(45)


119869120572(119908

119898

119908119899

) lt (1minus120582120572

119862120572

)

sdot 119903120572

[

[


119895=0119862119895+1120572

(120582120572

119862120572

)

119898+119895

+119862119899minus119898minus1120572

(120582120572

119862120572

)

119899minus2]

]

= (1minus120582120572

119862120572

)

sdot 119903120572

[

[

119862120572(

120582120572

119862120572

)

119898 119899minus119898minus2sum

119895=0120582119895

120572+(

119862120572

1205822120572

)120582119899

120572

119862119898

120572

]

]

(46)



119898isin0cupN 119908119898+1

isin119879 (119908119898

) (47)


120572le 120582

119898

120572


119898rarrinfin

sup119899gt119898

119869120572(119908

119898

119908119899


sup119899gt119898

(1minus120582120572

119862120572

)

sdot 119903120572[119862

120572(

120582120572

119862120572

)

119898

(1 minus 120582120572)minus1

+(119862120572

1205822120572

)120582119899

120572

119862119898

120572

]


(1minus120582120572

119862120572

) 119903120572[119862

120572(

120582120572

119862120572

)

119898

(1minus120582120572)minus1

+(119862120572

1205822120572

)(120582120572

119862120572

)

119898

]= 0

(48)


0isin 119883 by



119898rarrinfin

119869120572(119908 119908

119898

) = 0 (49)


= 119908119898 and 119908

119898= 119908 for 119898 isin N by



119898 isin N) and (119908119898



119898rarrinfin

119901120572(119908 119908

119898


119901120572(119908

119898 119906

119898) = 0 (50)



[119896]



119908(119898+1)119896

isin 119879[119896]

(119908119898119896


=

119908119898minus1+119896

119898 isin N) we see that (119909119898

119898 isin N) sub 119879[119896]

(119883)LIM119871minusP119862A


119898=

119908(119898+1)119896

119898 isin N) sub 119879[119896]

(119883) and (119906119898

= 119908119898119896

119898 isin N) sub

119879[119896]


119898isin 119879

[119896]


of (119909119898








[119896] is left P119862A-



such that 119908 isin 119879[119896]

(119908)



119862A)









119862A isin J119871(119883P119862A)


-


= 119863119871minusJ119862A120578119883120572

1198832



119906119908isin119883119863

119871minusJ119862A1119883120572 (119906 119908)

=max 119869120572(119906 119908) 119869

120572(119908 119906)


119906119908isin119883119863

119871minusJ119862A2119883120572 (119906 119908) = 119869

120572(119906 119908)

(51)





119909119910isin119883119862

120572sdot 119863

119871minusJ119862A120578119883120572

(119879 (119909) 119879 (119910))

le 120582120572119869120572(119909 119910)

(52)

(B) If J119862A isin J119877


D119877minusJ119862A119883120578


= 119863119877minusJ119862A120578119883120572

1198832

rarr



119906119908isin119883119863

119877minusJ119862A1119883120572 (119906 119908)

=max 119869120572(119906 119908) 119869

120572(119908 119906)


119906119908isin119883119863

119877minusJ119862A2119883120572 (119906 119908) = 119869

120572(119906 119908)

(53)




119909119910isin119883119862

120572sdot 119863

119877minusJ119862A120578119883120572

(119879 (119909) 119879 (119910))

le 120582120572119869120572(119909 119910)

(54)






(A) Given 1199080



119898

= 119879[119898]

(1199080)


(119908119898119898isin0cupN) = )

whenever


119898rarrinfin

sup119899gt119898

119869120572(119908

119898

119908119899

) = 0


119898rarrinfin

sup119899gt119898

119869120572(119908

119899

119908119898

) = 0)

(55)



isin 119883









[119896]

) is


(119909119898

119898 isin N) in 119879[119896]





119898 119898 isin N) and (119906


forall119898isinN V

119898= 119879

[119896]





119879[119896]

(119909) (119909 = 119879[119896]

(119909))


119862A)



120572120572isinA isin [0 1)A




119871minusJ119862A119883120578


119877minusJ119862A119883120578




(119908119898

= 119879[119898]



[119896]

) is left(right)P




119898

= 119879[119898]




120572(V 119879 (V)) = 119869

120572(119879 (V) V) = 0 (56)




) is left (right)P


Fix (119879[119896]

) = Fix (119879) = 119908 (57)


= 119879[119898]




120572(119908 119908) = 0 (58)




[119896]

(V) = 119879[2119896]

(V) 119879(V) =

119879[2119896]


0 lt 1198691205720

(V 119879 (V)) = 1198691205720

(119879[2119896]

(V) 119879[2119896](119879 (V)))

le 119863119871minusJ119862A1205781198831205720

(119879[2119896]

(V) 119879[2119896](119879 (V)))


le (1205821205720

1198621205720

)1198691205720

(119879[2119896minus1]

(V) 119879[2119896minus1](119879 (V)))

le (1205821205720

1198621205720

) sdot119863119871minusJ119862A1205781198831205720

(119879[2119896minus1]

(V) 119879[2119896minus1](119879 (V)))

le (1205821205720

1198621205720

)

2

1198691205720

(119879[2119896minus2]


le (1205821205720

1198621205720

)

2119896

1198691205720

(V 119879 (V)) lt 1198691205720

(V 119879 (V))

(59)



120572(V 119879 (V)) = 0 (60)




119879[119896]

(V) = 119879[2119896]


0 lt 1198691205720

(119879 (V) V) = 1198691205720

(119879[119896+1]

(V) 119879[2119896](V))

le

119896minus2sum

119898=1119862119898

12057201198691205720

(119879[119896+119898]

(V) 119879[119896+119898+1](V))

+119862119896minus21205720

1198691205720

(119879[2119896minus1]

(V) 119879[2119896](V))

le

119896minus2sum

119898=1119862119898

1205720sdot 119863

119871minusJ119862A1205781198831205720

(119879[119896+119898]

(V) 119879[119896+119898+1](V))

+119862119896minus21205720

sdot 119863119871minusJ119862A1205781198831205720

(119879[2119896minus1]

(V) 119879[2119896](V))

le

119896minus2sum

119898=1119862119898

1205720(

1205821205720

1198621205720

)

119896+119898

1198691205720

(V 119879 (V))

+119862119896minus21205720

(1205821205720

1198621205720

)

2119896minus1

1198691205720

(V 119879 (V)) = 0

(61)



120572(119879 (V) V) = 0 (62)






)


120572 120572 isin A is


120572(119879(V) V) gt



120572(119879(V) V) gt 0 or 119869

120572(V 119879(V)) gt 0 However by


120572(V V) = 0 In fact


120572(V V) le 119862

120572[119869

120572(V 119879(V))+119869

120572(119879(V) V)] = 0




1199011205720(119906 119908) gt 0 or 119901

1205720(119908 119906) gt 0 By applying


1198691205720(119906 119908) gt 0 or 119869

1205720(119908 119906) gt


exist1205720isinA

[1198691205720

(119906 119908) gt 0 1198691205720

(119906 119908)

= 1198691205720

(119879 (119906) 119879 (119908)) le119863119871minusJ119862A1205781198831205720

(119879 (119906) 119879 (119908))

le(1205821205720

1198621205720

)1198691205720

(119906 119908) lt 1198691205720

(119906 119908)] or [1198691205720

(119908 119906)

gt 0 1198691205720

(119908 119906) = 1198691205720

(119879 (119908) 119879 (119906))

le119863119871minusJ119862A1205781198831205720

(119879 (119908) 119879 (119906)) le(1205821205720

1198621205720

)1198691205720

(119908 119906)

lt 1198691205720

(119908 119906)]

(64)




rarr [0infin)

be of the form

119901 (119906 119908)

=

0 if 119906 ge 119908 119906 119908 cap (0 6) = 119906 119908

(119908 minus 119906)4 if 119906 lt 119908 119906 119908 cap (0 6) = 119906 119908

120574 if 119906 119908 cap (0 6) = 119906 119908

(65)


space In fact





8(119908 minus V)4 = 8[119901(119906 V) + 119901(V 119908)]


(119908 minus 119906)4 and 119891(V0) = min



and V0 = (119906 + 119908)2


119906119908isin(06)119906lt119908(119908 minus 119906)4

=


119906leVle1199088[119901(119906 V) + 119901(V 119908)] =

sup119906119908isin(06)119906lt119908min


(6 minus 3)4] = 8[81 + 81] = 1296




4lt

8(V minus 119906)4= 8[119901(119906 V) + 119901(V 119908)]


1296 le 8[119901(119906 V) + 119901(V 119908)] = 8[120574 + 120574]


8120574 le 8[119901(119906 V) + 119901(V 119908)]



119901(119906119908) = 119901(119908 119906) does not hold




119898= 3





(119906119898 119898isinN)= [0 3]



119901 (119906 119908) =

0 if 119860 cap 119906 119908 = 119906 119908

120574 if 119860 cap 119906 119908 = 119906 119908

(67)









119901(119906 119906) = 0 does not hold(3)P


(119906119898 119898isinN)= LIM119877minusP11

(119906119898 119898isinN)= 119860 for





119901 (119906 119908) =

0 if 119906 ge 119908

(119908 minus 119906)3 if 119906 lt 119908

(68)






4119901(V 119908) = 4[119901(119906 V) + 119901(V 119908)]


119891(V0) = min119906⩽V⩽119908119891(V) = (119908 minus 119906)

3 where V0 = (119906 + 119908)2 is


4(119908 minus 119906)[1199082+ 119908119906 + 119906

2+ 3V2 minus 3V(119908 + 119906)]



3lt 4(V minus 119906)

3= 4119901(119906 V) = 4[119901(119906

V) + 119901(V 119908)]



119901(119906119908) = 119901(119908 119906) does not hold




(119906119898 119898isinN)= [2 6] and

LIM119877minusP41(119906119898 119898isinN)

= [1 2] for sequence (119906119898



1198832


119901 (119906 119908) =

0 if 119906 ge 119908

(119906 minus 119908)2 if 119906 lt 119908

(69)

Let

119864 = [0 3) cup (3 6] (70)


2rarr [0infin) is

of the form

119869 (119906 119908) =

119901 (119906 119908) if 119864 cap 119906 119908 = 119906 119908

120583 if 119864 cap 119906 119908 = 119906 119908

(71)


36 and 119869(6 0) = 0(2)J

21 = 119869 isin J119871(119883P21)

capJ119877(119883P21)

SeeTheorem 14




119901 (119906 119908)

=

0 if 119906 ge 119908 119906 119908 cap (0 6) = 119906 119908

(119908 minus 119906)4 if 119906 lt 119908 119906 119908 cap (0 6) = 119906 119908

120574 if 119906 119908 cap (0 6) = 119906 119908

(72)


119879 (119906) =

[1 2] if 119906 isin [0 3) cup (4 6]

(4 6) if 119906 isin [3 4] (73)


Let119864 = [0 3) cup (4 6] (74)


119869 (119906 119908) =

119901 (119906 119908) if 119864 cap 119906 119908 = 119906 119908

120574 if 119864 cap 119906 119908 = 119906 119908

(75)

(1) (119883P81) where P




(2) (119883 119879) is a (119863 = D119871minusJ8112119883 = D

119877minusJ8112119883 120582 isin [2048


119909119910isin1198838 sdot

119863(119879(119909) 119879(119910)) le 120582119869(119909 119910) where

119863 (119880119882) = maxsup119906isin119880

119869 (119906119882) sup119908isin119882

119869 (119880 119908)

119880119882 isin 2119883

(76)



119864 and sup119906isin119880

inf119908isin119880

119869(119906 119908) = sup119906isin119880

119869(119906 119906) = 119901(119906 119906) =

0 = 0 Thus 4119863(119879(119909) 119879(119910)) = 0 le 120582119869(119909 119910)



inf119908isin119882

119869(119906 119908) =

sup119906isin119880

inf119908isin119882

(119908 minus 119906)4 = sup

119906isin119880(4 minus 119906)

4= 81 and

sup119908isin119882

inf119906isin119880

119869(119906 119908) = sup119908isin119882

inf119906isin119880

(119908 minus 119906)4 =

sup119908isin119882





119869(119906 119908) = sup119906isin119880

inf119908isin119882

119901(119906 119908) =

sup119908isin119882

inf119906isin119880

119869(119906 119908) = sup119908isin119882

inf119906isin119880

119901(119906 119908) = 0Therefore 8119863(119879(119909) 119879(119910)) = 0 le 120582119869(119909 119910)




exist119910isin119879(119909)

119869(119909




120573isin(0infin)119869(1199090

1199100) lt 119869(1199090 119879(1199090)) + 120573



120573isin(0infin)119869(1199090 1199100) lt 119869(1199090 119879(1199090)) + 120573


120573isin(0infin)119869(1199090 1199100) lt

119869(1199090 119879(1199090)) + 120573



119869(1199090 1199100) lt 119869(1199090 119879(1199090)) +

120573


120573isin(0infin)119869(1199090 1199100) lt 119869(1199090 119879(1199090)) + 120573


1199080isin 119883 In fact if1199080

isin 119883 and (119908119898


119898+1isin 119879(119908

119898


sup119899gt119898

119869(119908119898

119908119899

) =


sup119899gt119898

119869(119908119899

119908119898

) = 0) then forall119898ge2 119908

119898



119901(119908119908119898

) =


119898rarrinfin119901(119908

119898



119869(119908 119908119898

) =


119898rarrinfin119869(119908

119898





(119909119898 119898isinN)= (LIM119877minusP81

(119909119898 119898isinN)= ))


119898 isin N) and (119906119898

119898 isin N)


119898isin 119879(119906

119898) Then forall

119898ge2 119909119898

isin [1 2] 2 isin





following


0isin 119883




0isin 119883





119909119910isin1198838119863(119879(119909)


119901(119906119882) sup



119909119910isin1198838119863(119879(119909) 119879(119910)) le 1205820119901(119909 119910) Letting 1199090 = 2


[1 2] = 119880 119879(1199100) = (4 6) = 119882 sup119906isin[12]119901(119906 (4 6)) =

sup119906isin[12](4 minus 119906)

4= 34 = 81 and sup

119908isin(46)119901([1 2] 119908) =




119862A)


119862A whereJ



119862A)whenJ119862A = P


119862A





119860 = 1198601 cup1198602 1198601 = (0 2] 1198602 = [4 6) (77)

Let 119901 1198832


119901 (119906 119908) =

0 if 119860 cap 119906 119908 = 119906 119908

120574 if 119860 cap 119906 119908 = 119906 119908

(78)

and letJ11 = P


119879 (119906) =

1198602 for 119906 isin (0 3)

119860 for 119906 = 3

1198601 for 119906 isin (3 6)

(79)


Section 11(2) (119883 119879) is a (D



119909119910isin119883119863

119871minusP1112119883 (119879(119909) 119879(119910)) le 120582119901(119909


and maxsup119906isin119879(119909)

119901(119906 119879(119910)) sup119908isin119879(119910)



exist119910isin119879(119909)



119901 (1199090 1199100) = 119901 (1199090 119879 (1199090))

=

0 if 1199090 isin 1198601

120574 for 1199090 isin (0 3) 1198601

(80)

Therefore 119901(1199090 1199100) lt 119901(1199090 119879(1199090)) + 120573



119901 (1199090 1199100) = 119901 (1199090 119879 (1199090))

=

0 if 1199090 isin 1198602

120574 for 1199090 isin (3 6) 1198602

(81)







sup119899gt119898

119901(119908119898

119908119899


lim119898rarrinfin

119901(119908119908119898


(79)forall119898⩾1 119908

119898

isin 119879(119908119898minus1



(119908119898119898isin0cupN)




119898 isin N) sub 119879[2]

(119883) = 119860



119898 isin N) and (119906119898


119898isin 119879(119906


exist1198980isinN

forall119898⩾1198980

119909119898



and Fix(119879[2]) = 119860

(6) For (119883P11)P11 = 119901 J

11 = P11 and



isin

119883 1199081isin 119879(119908

0) and 119908

2= 119906 isin 119879(119908


and forall119898⩾3 119908

119898

= 119906 then sequence (119908119898




1198832


119901 (119906 119908) =

0 if 119906 ge 119908

(119906 minus 119908)2 if 119906 lt 119908

(82)


119879 (119906) = ([0 3) cup (3 6]) 119906 for 119906 isin [0 6] (83)

Let

119864 = [0 3) cup (3 6] (84)


2rarr [0infin) is

of the form

119869 (119906 119908) =

119901 (119906 119908) if 119864 cap 119906 119908 = 119906 119908

120583 if 119864 cap 119906 119908 = 119906 119908

(85)


119869(0 6) = 36 and 119869(6 0) = 0(2)J

21 = 119869 isin J119871(119883P21)

capJ119877(119883P21)

SeeTheorem 14




[0 1) and

119863(119880119882) = maxsup119906isin119880

119869 (119906119882) sup119908isin119882

119869 (119880 119908)

119880119882 isin 2119883

(86)




119906

cup 119882119906and

inf119908isin119882

119869 (119906 119908)

le

inf119908isin119882119906119902 (119906 119908) = 0 if 119882

119906

= 119908 isin 119882 119906 ge 119908 =

inf119908isin119882119906

(119906 minus 119908)2= 0 if 119882

119906= 119908 isin 119882 119906 lt 119908 =

(87)


cup 119880119908and

inf119906isin119880

119869 (119906 119908)

le

inf119906isin119880119908119902 (119906 119908) = 0 if 119880

119906

= 119906 isin 119880 119906 ge 119908 =

inf119906isin119880119908

(119906 minus 119908)2= 0 if 119880

119906= 119906 isin 119880 119906 lt 119908 =

(88)

By (86) 2119863(119879(119909) 119879(119910)) = 0 le 120582119869(119909 119910)

Case 2 If 119909 = 119910 = 3 then 119869(119909 119910) = 120583 and 119879(119909) =


2119863(119880119880) = 0 le 120582119869(119909 119910)



inf119908isin119882

119869(119906 119908) =


119869(119906 119908) =


inf119906isin119880


119908

cup 119880119908and

inf119906isin119880

119869 (119906 119908)

le

inf119906isin119880119908119902 (119906 119908) = 0 if 119880

119906

= 119906 isin 119880 119906 ge 119908 =

inf119906isin119880119908

(119906 minus 119908)2= 0 if 119880

119906= 119906 isin 119880 119906 lt 119908 =

(89)

Thus 2119863(119879(119909) 119879(119910)) = 0 ⩽ 120582119869(119909 119910)


(3 6]) 119910 = 119882 sub 119864 and sup119906isin119880

inf119908isin119882

119869(119906 119908) = 0 sincefor 119906 isin 119880

inf119908isin119882

119869 (119906 119908)

le

inf119908isin119882119906119902 (119906 119908) = 0 if 119882

119906

= 119908 isin 119882 119906 ge 119908 =

inf119908isin119882119906

(119906 minus 119908)2= 0 if 119882

119906= 119908 isin 119882 119906 lt 119908 =

(90)


inf119906isin119880

119869(119906 119908) = 0 since inf119906isin119880

119869(119906 119908) = 119869(119908119908) = 0 for 119908 isin 119882 Thus 2119863(119879(119909) 119879(119910)) = 0 le 120582119869(119909 119910)



exist119910isin119879(119909)

119869(119909




inf119908isin119882

119869(1199090 119908) = 0 since

inf119908isin119882

119869 (1199090 119908)

le

inf119908isin1198821199090119902 (1199090 119908) = 0 if 119882

1199090 = 119908 isin 119882 1199090 ge 119908 =

inf119908isin1198821199090

(1199090 minus 119908)2= 0 if 119882

1199090= 119908 isin 119882 1199090 lt 119908 =

(91)



12)

then 119869(1199090 1199100) lt 119869(1199090 119879(1199090)) + 120574



119869(1199090 1199100) lt 119869(1199090 119879(1199090)) + 120574




119869(1199090 119879(1199090)) + 120574


1199080


119898



sup119899gt119898

119869(119908119898

119908119899


lim119898rarrinfin

119869(119908 119908119898



1isin 119879(119908

0) = ([0 3) cup

(3 6]) 1199080 and forall

119898ge2 119908119898

isin 119879(119908119898minus1


(119908119898119898isin0cupN)

Case 2 If 1199080= 3 then 119908

1isin 119879(119908

0) = [0 3) cup (3 6] 1199082

isin

119879(1199081) = ([0 3) cup (3 6]) 119908

1 and forall

119898⩾3 119908119898

isin 119879(119908119898minus1

) sub




0isin 119883 and



119898rarrinfinsup

119899gt119898119869(119908

119898

119908119899

) =

0 of (119908119898


[2]) is a left P


119898 119898 isin N) sub 119879

[2](119883) =



119898 isin N) and (119906119898

119898 isin N)


119898isin 119879(119906


exist1198980isinN

forall119898ge1198980

119909119898


(119909119898 119898isinN)and 6 isin 119879

[2](6)

(7) ForP21 = 119901J



that 6 isin 119879[2]


0isin 119883






11 = 119901


119879 (119906) =

4 for 119906 isin (0 3)

2 for 119906 isin [3 6) (92)



119909119910isin119883119863

119871minusP111119883 (119879(119909) 119879(119910)) =

max119901(119879(119909) 119879(119910)) 119901(119879(119910) 119879(119909)) le 120582119901(119909 119910) and Fix(119879) =






119898+1= 119879(119908

119898


sup119899gt119898

119901(119908119898

119908119899

) =


119898



(119908119898119898isin0cupN)




119898 isin N) sub

119879[2]



119898 isin N) and (119906119898

119898 isin N)


119898= 119879

[2](119906

119898) then by (77) (78) and (92)



(2) isin

LIM119871minusP11(119909119898 119898isinN)

and 4 = 119879[2]





119883119860 then 119901(119906 119908) = 119901(119908 119906) = 120574 gt 0(6) For P

11 = 119901 (119883 119879) and J11 = P





119889


119889



(119883 119889) rarr (CB(119883)119867119889











References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(A) LetJ119862A isin J119871

(119883P119862A) If


119880119882isin2119883 119863119871minusJ119862A

12119883120572 (119880119882)


119869120572(119906119882) sup

119908isin119882

119869120572(119880 119908)


119880119882isin2119883 119863119871minusJ119862A

22119883120572 (119880119882)


119869120572(119906119882) sup

119908isin119882

119869120572(119908 119880)


119880119882isin2119883 119863119871minusJ119862A

32119883120572 (119880119882) = sup119906isin119880

119869120572(119906119882)

(22)


1205782119883 = 119863119871minusJ119862A





119909119910isin119883119862

120572sdot 119863

119871minusJ119862A

1205782119883120572 (119879 (119909) 119879 (119910))

le 120582120572119869120572(119909 119910)

(23)


1205782119883 120582)-quasi-119888119900119899119905119903119886119888119905119894119900119899on 119883

(B) LetJ119862A isin J119877

(119883P119862A) If


119880119882isin2119883 119863119877minusJ119862A

12119883120572 (119880119882)


119869120572(119906119882) sup

119908isin119882

119869120572(119880 119908)


119880119882isin2119883 119863119877minusJ119862A

22119883120572 (119880119882)


119869120572(119906119882) sup

119908isin119882

119869120572(119908 119880)


119880119882isin2119883 119863119877minusJ119862A

32119883120572 (119880119882) = sup119906isin119880

119869120572(119906119882)

(24)


1205782119883 = 119863119877minusJ119862A





119909119910isin119883119862

120572sdot 119863

119877minusJ119862A

1205782119883120572 (119879 (119909) 119879 (119910))

le 120582120572119869120572(119909 119910)

(25)




1205782119883 and D119877minusJ119862A




1205782119883



119877minusJ119862A






120572120572isinA isin [0 1)A




119871minusJ119862A


(D119877minusJ119862A





A



120572(119909 119910) lt 119869

120572(119909 119879 (119909)) + 120573

120572 (26)


120572(119910 119909) lt 119869

120572(119879 (119909) 119909) + 120573

120572) (27)


119898



119898+1isin 119879(119908

119898




[119896]

) is left (right)P




119898isin0cupN 119908119898+1

isin 119879(119908119898


) such that (119908119898






0isin 119883 and (119883 119879

[119896]




120572 119883


choose

119903 = 119903120572120572isinA

isin (0infin)A (28)


such that


120572(119908

0 119879 (119908

0)) lt(1minus

120582120572

119862120572

) 119903120572 (29)

Put


(0)120572

=(1minus120582120572

119862120572

) 119903120572minus 119869

120572(119908

0 119879 (119908

0)) (30)


= 120573(0)120572

120572isinA isin


1isin 119879(119908

0) such that


120572(119908

0 119908

1) lt 119869

120572(119908

0 119879 (119908

0)) + 120573

(0)120572

(31)



120572(119908

0 119908

1) lt(1minus

120582120572

119862120572

) 119903120572 (32)

Put now


(1)120572

=(120582120572

119862120572

)[(1minus120582120572

119862120572

) 119903120572minus 119869

120572(119908

0 119908

1)] (33)


(1)120572



2isin 119879(119908

1) such that


120572(119908

1 119908

2) lt 119869

120572(119908

1 119879 (119908

1)) + 120573

(1)120572

(34)

Observe that


120572(119908

1 119908

2) lt(

120582120572

119862120572

)(1minus120582120572

119862120572

) 119903120572 (35)



120572(119908

1 119908

2) lt 119869

120572(119908

1 119879 (119908

1)) + 120573

(1)120572

le sup 119869120572(119906 119879 (119908

1)) 119906 isin119879 (119908

0) + 120573

(1)120572

le119863119871minusJ119862A

1205782119883120572 (119879 (1199080) 119879 (119908

1)) + 120573

(1)120572

le(120582120572

119862120572

) 119869120572(119908

0 119908

1) + 120573

(1)120572

=(120582120572

119862120572

)(1minus120582120572

119862120572

) 119903120572

(36)



(2)120572

=(120582120572

119862120572

)[(120582120572

119862120572

)(1minus120582120572

119862120572

) 119903120572minus 119869

120572(119908

1 119908

2)]

(37)

Then in view of (35) 120573(2)= 120573

(2)120572



3isin

119879(1199082) such that


120572(119908

2 119908

3) lt 119869

120572(119908

2 119879 (119908

2)) + 120573

(2)120572

(38)



120572(119908

2 119908

3) lt(

120582120572

119862120572

)

2

(1minus120582120572

119862120572

) 119903120572 (39)



120572(119908

2 119908

3) lt 119869

120572(119908

2 119879 (119908

2)) + 120573

(2)120572

⩽ sup119906isin119879(119908

1)

119869120572(119906 119879 (119908

2)) + 120573

(2)120572

le119863119871minusJ119862A

1205782119883120572 (119879 (1199081) 119879 (119908

2)) + 120573

(2)120572

le(120582120572

119862120572

) 119869120572(119908

1 119908

2) + 120573

(2)120572

=(120582120572

119862120572

)

2

(1minus120582120572

119862120572

) 119903120572

(40)





119898+1isin119879 (119908

119898

) (41)


(119898)

=

120573(119898)

120572120572isinA

where


119898isinN 120573(119898)

120572=(

120582120572

119862120572

)

sdot [(120582120572

119862120572

)

119898minus1(1minus

120582120572

119862120572

) 119903120572minus 119869

120572(119908

119898minus1 119908

119898

)]

(42)


(119898)

isin (0infin)A


119898isinN 119869120572(119908

119898

119908119898+1

) lt 119869120572(119908

119898

119879 (119908119898

))

+ 120573(119898)

120572

(43)


119898isinN 119869120572(119908

119898

119908119898+1

)

lt(120582120572

119862120572

)

119898

(1minus120582120572

119862120572

) 119903120572

(44)



forall120572isinA

119869120572(119908

119898

119908119899

) ⩽119862120572119869120572(119908

119898

119908119898+1

)

+1198622120572119869120572(119908

119898+1 119908


+ 119862119899minus119898minus1120572

119869120572(119908

119899minus2 119908

119899minus1) +119862

119899minus119898minus1120572

119869120572(119908

119899minus1 119908

119899

)

=


119895=0119862119895+1120572

119869120572(119908

119898+119895

119908119898+119895+1

)

+119862119899minus119898minus1120572

119869120572(119908

119899minus1 119908

119899

)

(45)


119869120572(119908

119898

119908119899

) lt (1minus120582120572

119862120572

)

sdot 119903120572

[

[


119895=0119862119895+1120572

(120582120572

119862120572

)

119898+119895

+119862119899minus119898minus1120572

(120582120572

119862120572

)

119899minus2]

]

= (1minus120582120572

119862120572

)

sdot 119903120572

[

[

119862120572(

120582120572

119862120572

)

119898 119899minus119898minus2sum

119895=0120582119895

120572+(

119862120572

1205822120572

)120582119899

120572

119862119898

120572

]

]

(46)



119898isin0cupN 119908119898+1

isin119879 (119908119898

) (47)


120572le 120582

119898

120572


119898rarrinfin

sup119899gt119898

119869120572(119908

119898

119908119899


sup119899gt119898

(1minus120582120572

119862120572

)

sdot 119903120572[119862

120572(

120582120572

119862120572

)

119898

(1 minus 120582120572)minus1

+(119862120572

1205822120572

)120582119899

120572

119862119898

120572

]


(1minus120582120572

119862120572

) 119903120572[119862

120572(

120582120572

119862120572

)

119898

(1minus120582120572)minus1

+(119862120572

1205822120572

)(120582120572

119862120572

)

119898

]= 0

(48)


0isin 119883 by



119898rarrinfin

119869120572(119908 119908

119898

) = 0 (49)


= 119908119898 and 119908

119898= 119908 for 119898 isin N by



119898 isin N) and (119908119898



119898rarrinfin

119901120572(119908 119908

119898


119901120572(119908

119898 119906

119898) = 0 (50)



[119896]



119908(119898+1)119896

isin 119879[119896]

(119908119898119896


=

119908119898minus1+119896

119898 isin N) we see that (119909119898

119898 isin N) sub 119879[119896]

(119883)LIM119871minusP119862A


119898=

119908(119898+1)119896

119898 isin N) sub 119879[119896]

(119883) and (119906119898

= 119908119898119896

119898 isin N) sub

119879[119896]


119898isin 119879

[119896]


of (119909119898








[119896] is left P119862A-



such that 119908 isin 119879[119896]

(119908)



119862A)









119862A isin J119871(119883P119862A)


-


= 119863119871minusJ119862A120578119883120572

1198832



119906119908isin119883119863

119871minusJ119862A1119883120572 (119906 119908)

=max 119869120572(119906 119908) 119869

120572(119908 119906)


119906119908isin119883119863

119871minusJ119862A2119883120572 (119906 119908) = 119869

120572(119906 119908)

(51)





119909119910isin119883119862

120572sdot 119863

119871minusJ119862A120578119883120572

(119879 (119909) 119879 (119910))

le 120582120572119869120572(119909 119910)

(52)

(B) If J119862A isin J119877


D119877minusJ119862A119883120578


= 119863119877minusJ119862A120578119883120572

1198832

rarr



119906119908isin119883119863

119877minusJ119862A1119883120572 (119906 119908)

=max 119869120572(119906 119908) 119869

120572(119908 119906)


119906119908isin119883119863

119877minusJ119862A2119883120572 (119906 119908) = 119869

120572(119906 119908)

(53)




119909119910isin119883119862

120572sdot 119863

119877minusJ119862A120578119883120572

(119879 (119909) 119879 (119910))

le 120582120572119869120572(119909 119910)

(54)






(A) Given 1199080



119898

= 119879[119898]

(1199080)


(119908119898119898isin0cupN) = )

whenever


119898rarrinfin

sup119899gt119898

119869120572(119908

119898

119908119899

) = 0


119898rarrinfin

sup119899gt119898

119869120572(119908

119899

119908119898

) = 0)

(55)



isin 119883









[119896]

) is


(119909119898

119898 isin N) in 119879[119896]





119898 119898 isin N) and (119906


forall119898isinN V

119898= 119879

[119896]





119879[119896]

(119909) (119909 = 119879[119896]

(119909))


119862A)



120572120572isinA isin [0 1)A




119871minusJ119862A119883120578


119877minusJ119862A119883120578




(119908119898

= 119879[119898]



[119896]

) is left(right)P




119898

= 119879[119898]




120572(V 119879 (V)) = 119869

120572(119879 (V) V) = 0 (56)




) is left (right)P


Fix (119879[119896]

) = Fix (119879) = 119908 (57)


= 119879[119898]




120572(119908 119908) = 0 (58)




[119896]

(V) = 119879[2119896]

(V) 119879(V) =

119879[2119896]


0 lt 1198691205720

(V 119879 (V)) = 1198691205720

(119879[2119896]

(V) 119879[2119896](119879 (V)))

le 119863119871minusJ119862A1205781198831205720

(119879[2119896]

(V) 119879[2119896](119879 (V)))


le (1205821205720

1198621205720

)1198691205720

(119879[2119896minus1]

(V) 119879[2119896minus1](119879 (V)))

le (1205821205720

1198621205720

) sdot119863119871minusJ119862A1205781198831205720

(119879[2119896minus1]

(V) 119879[2119896minus1](119879 (V)))

le (1205821205720

1198621205720

)

2

1198691205720

(119879[2119896minus2]


le (1205821205720

1198621205720

)

2119896

1198691205720

(V 119879 (V)) lt 1198691205720

(V 119879 (V))

(59)



120572(V 119879 (V)) = 0 (60)




119879[119896]

(V) = 119879[2119896]


0 lt 1198691205720

(119879 (V) V) = 1198691205720

(119879[119896+1]

(V) 119879[2119896](V))

le

119896minus2sum

119898=1119862119898

12057201198691205720

(119879[119896+119898]

(V) 119879[119896+119898+1](V))

+119862119896minus21205720

1198691205720

(119879[2119896minus1]

(V) 119879[2119896](V))

le

119896minus2sum

119898=1119862119898

1205720sdot 119863

119871minusJ119862A1205781198831205720

(119879[119896+119898]

(V) 119879[119896+119898+1](V))

+119862119896minus21205720

sdot 119863119871minusJ119862A1205781198831205720

(119879[2119896minus1]

(V) 119879[2119896](V))

le

119896minus2sum

119898=1119862119898

1205720(

1205821205720

1198621205720

)

119896+119898

1198691205720

(V 119879 (V))

+119862119896minus21205720

(1205821205720

1198621205720

)

2119896minus1

1198691205720

(V 119879 (V)) = 0

(61)



120572(119879 (V) V) = 0 (62)






)


120572 120572 isin A is


120572(119879(V) V) gt



120572(119879(V) V) gt 0 or 119869

120572(V 119879(V)) gt 0 However by


120572(V V) = 0 In fact


120572(V V) le 119862

120572[119869

120572(V 119879(V))+119869

120572(119879(V) V)] = 0




1199011205720(119906 119908) gt 0 or 119901

1205720(119908 119906) gt 0 By applying


1198691205720(119906 119908) gt 0 or 119869

1205720(119908 119906) gt


exist1205720isinA

[1198691205720

(119906 119908) gt 0 1198691205720

(119906 119908)

= 1198691205720

(119879 (119906) 119879 (119908)) le119863119871minusJ119862A1205781198831205720

(119879 (119906) 119879 (119908))

le(1205821205720

1198621205720

)1198691205720

(119906 119908) lt 1198691205720

(119906 119908)] or [1198691205720

(119908 119906)

gt 0 1198691205720

(119908 119906) = 1198691205720

(119879 (119908) 119879 (119906))

le119863119871minusJ119862A1205781198831205720

(119879 (119908) 119879 (119906)) le(1205821205720

1198621205720

)1198691205720

(119908 119906)

lt 1198691205720

(119908 119906)]

(64)




rarr [0infin)

be of the form

119901 (119906 119908)

=

0 if 119906 ge 119908 119906 119908 cap (0 6) = 119906 119908

(119908 minus 119906)4 if 119906 lt 119908 119906 119908 cap (0 6) = 119906 119908

120574 if 119906 119908 cap (0 6) = 119906 119908

(65)


space In fact





8(119908 minus V)4 = 8[119901(119906 V) + 119901(V 119908)]


(119908 minus 119906)4 and 119891(V0) = min



and V0 = (119906 + 119908)2


119906119908isin(06)119906lt119908(119908 minus 119906)4

=


119906leVle1199088[119901(119906 V) + 119901(V 119908)] =

sup119906119908isin(06)119906lt119908min


(6 minus 3)4] = 8[81 + 81] = 1296




4lt

8(V minus 119906)4= 8[119901(119906 V) + 119901(V 119908)]


1296 le 8[119901(119906 V) + 119901(V 119908)] = 8[120574 + 120574]


8120574 le 8[119901(119906 V) + 119901(V 119908)]



119901(119906119908) = 119901(119908 119906) does not hold




119898= 3





(119906119898 119898isinN)= [0 3]



119901 (119906 119908) =

0 if 119860 cap 119906 119908 = 119906 119908

120574 if 119860 cap 119906 119908 = 119906 119908

(67)









119901(119906 119906) = 0 does not hold(3)P


(119906119898 119898isinN)= LIM119877minusP11

(119906119898 119898isinN)= 119860 for





119901 (119906 119908) =

0 if 119906 ge 119908

(119908 minus 119906)3 if 119906 lt 119908

(68)






4119901(V 119908) = 4[119901(119906 V) + 119901(V 119908)]


119891(V0) = min119906⩽V⩽119908119891(V) = (119908 minus 119906)

3 where V0 = (119906 + 119908)2 is


4(119908 minus 119906)[1199082+ 119908119906 + 119906

2+ 3V2 minus 3V(119908 + 119906)]



3lt 4(V minus 119906)

3= 4119901(119906 V) = 4[119901(119906

V) + 119901(V 119908)]



119901(119906119908) = 119901(119908 119906) does not hold




(119906119898 119898isinN)= [2 6] and

LIM119877minusP41(119906119898 119898isinN)

= [1 2] for sequence (119906119898



1198832


119901 (119906 119908) =

0 if 119906 ge 119908

(119906 minus 119908)2 if 119906 lt 119908

(69)

Let

119864 = [0 3) cup (3 6] (70)


2rarr [0infin) is

of the form

119869 (119906 119908) =

119901 (119906 119908) if 119864 cap 119906 119908 = 119906 119908

120583 if 119864 cap 119906 119908 = 119906 119908

(71)


36 and 119869(6 0) = 0(2)J

21 = 119869 isin J119871(119883P21)

capJ119877(119883P21)

SeeTheorem 14




119901 (119906 119908)

=

0 if 119906 ge 119908 119906 119908 cap (0 6) = 119906 119908

(119908 minus 119906)4 if 119906 lt 119908 119906 119908 cap (0 6) = 119906 119908

120574 if 119906 119908 cap (0 6) = 119906 119908

(72)


119879 (119906) =

[1 2] if 119906 isin [0 3) cup (4 6]

(4 6) if 119906 isin [3 4] (73)


Let119864 = [0 3) cup (4 6] (74)


119869 (119906 119908) =

119901 (119906 119908) if 119864 cap 119906 119908 = 119906 119908

120574 if 119864 cap 119906 119908 = 119906 119908

(75)

(1) (119883P81) where P




(2) (119883 119879) is a (119863 = D119871minusJ8112119883 = D

119877minusJ8112119883 120582 isin [2048


119909119910isin1198838 sdot

119863(119879(119909) 119879(119910)) le 120582119869(119909 119910) where

119863 (119880119882) = maxsup119906isin119880

119869 (119906119882) sup119908isin119882

119869 (119880 119908)

119880119882 isin 2119883

(76)



119864 and sup119906isin119880

inf119908isin119880

119869(119906 119908) = sup119906isin119880

119869(119906 119906) = 119901(119906 119906) =

0 = 0 Thus 4119863(119879(119909) 119879(119910)) = 0 le 120582119869(119909 119910)



inf119908isin119882

119869(119906 119908) =

sup119906isin119880

inf119908isin119882

(119908 minus 119906)4 = sup

119906isin119880(4 minus 119906)

4= 81 and

sup119908isin119882

inf119906isin119880

119869(119906 119908) = sup119908isin119882

inf119906isin119880

(119908 minus 119906)4 =

sup119908isin119882





119869(119906 119908) = sup119906isin119880

inf119908isin119882

119901(119906 119908) =

sup119908isin119882

inf119906isin119880

119869(119906 119908) = sup119908isin119882

inf119906isin119880

119901(119906 119908) = 0Therefore 8119863(119879(119909) 119879(119910)) = 0 le 120582119869(119909 119910)




exist119910isin119879(119909)

119869(119909




120573isin(0infin)119869(1199090

1199100) lt 119869(1199090 119879(1199090)) + 120573



120573isin(0infin)119869(1199090 1199100) lt 119869(1199090 119879(1199090)) + 120573


120573isin(0infin)119869(1199090 1199100) lt

119869(1199090 119879(1199090)) + 120573



119869(1199090 1199100) lt 119869(1199090 119879(1199090)) +

120573


120573isin(0infin)119869(1199090 1199100) lt 119869(1199090 119879(1199090)) + 120573


1199080isin 119883 In fact if1199080

isin 119883 and (119908119898


119898+1isin 119879(119908

119898


sup119899gt119898

119869(119908119898

119908119899

) =


sup119899gt119898

119869(119908119899

119908119898

) = 0) then forall119898ge2 119908

119898



119901(119908119908119898

) =


119898rarrinfin119901(119908

119898



119869(119908 119908119898

) =


119898rarrinfin119869(119908

119898





(119909119898 119898isinN)= (LIM119877minusP81

(119909119898 119898isinN)= ))


119898 isin N) and (119906119898

119898 isin N)


119898isin 119879(119906

119898) Then forall

119898ge2 119909119898

isin [1 2] 2 isin





following


0isin 119883




0isin 119883





119909119910isin1198838119863(119879(119909)


119901(119906119882) sup



119909119910isin1198838119863(119879(119909) 119879(119910)) le 1205820119901(119909 119910) Letting 1199090 = 2


[1 2] = 119880 119879(1199100) = (4 6) = 119882 sup119906isin[12]119901(119906 (4 6)) =

sup119906isin[12](4 minus 119906)

4= 34 = 81 and sup

119908isin(46)119901([1 2] 119908) =




119862A)


119862A whereJ



119862A)whenJ119862A = P


119862A





119860 = 1198601 cup1198602 1198601 = (0 2] 1198602 = [4 6) (77)

Let 119901 1198832


119901 (119906 119908) =

0 if 119860 cap 119906 119908 = 119906 119908

120574 if 119860 cap 119906 119908 = 119906 119908

(78)

and letJ11 = P


119879 (119906) =

1198602 for 119906 isin (0 3)

119860 for 119906 = 3

1198601 for 119906 isin (3 6)

(79)


Section 11(2) (119883 119879) is a (D



119909119910isin119883119863

119871minusP1112119883 (119879(119909) 119879(119910)) le 120582119901(119909


and maxsup119906isin119879(119909)

119901(119906 119879(119910)) sup119908isin119879(119910)



exist119910isin119879(119909)



119901 (1199090 1199100) = 119901 (1199090 119879 (1199090))

=

0 if 1199090 isin 1198601

120574 for 1199090 isin (0 3) 1198601

(80)

Therefore 119901(1199090 1199100) lt 119901(1199090 119879(1199090)) + 120573



119901 (1199090 1199100) = 119901 (1199090 119879 (1199090))

=

0 if 1199090 isin 1198602

120574 for 1199090 isin (3 6) 1198602

(81)







sup119899gt119898

119901(119908119898

119908119899


lim119898rarrinfin

119901(119908119908119898


(79)forall119898⩾1 119908

119898

isin 119879(119908119898minus1



(119908119898119898isin0cupN)




119898 isin N) sub 119879[2]

(119883) = 119860



119898 isin N) and (119906119898


119898isin 119879(119906


exist1198980isinN

forall119898⩾1198980

119909119898



and Fix(119879[2]) = 119860

(6) For (119883P11)P11 = 119901 J

11 = P11 and



isin

119883 1199081isin 119879(119908

0) and 119908

2= 119906 isin 119879(119908


and forall119898⩾3 119908

119898

= 119906 then sequence (119908119898




1198832


119901 (119906 119908) =

0 if 119906 ge 119908

(119906 minus 119908)2 if 119906 lt 119908

(82)


119879 (119906) = ([0 3) cup (3 6]) 119906 for 119906 isin [0 6] (83)

Let

119864 = [0 3) cup (3 6] (84)


2rarr [0infin) is

of the form

119869 (119906 119908) =

119901 (119906 119908) if 119864 cap 119906 119908 = 119906 119908

120583 if 119864 cap 119906 119908 = 119906 119908

(85)


119869(0 6) = 36 and 119869(6 0) = 0(2)J

21 = 119869 isin J119871(119883P21)

capJ119877(119883P21)

SeeTheorem 14




[0 1) and

119863(119880119882) = maxsup119906isin119880

119869 (119906119882) sup119908isin119882

119869 (119880 119908)

119880119882 isin 2119883

(86)




119906

cup 119882119906and

inf119908isin119882

119869 (119906 119908)

le

inf119908isin119882119906119902 (119906 119908) = 0 if 119882

119906

= 119908 isin 119882 119906 ge 119908 =

inf119908isin119882119906

(119906 minus 119908)2= 0 if 119882

119906= 119908 isin 119882 119906 lt 119908 =

(87)


cup 119880119908and

inf119906isin119880

119869 (119906 119908)

le

inf119906isin119880119908119902 (119906 119908) = 0 if 119880

119906

= 119906 isin 119880 119906 ge 119908 =

inf119906isin119880119908

(119906 minus 119908)2= 0 if 119880

119906= 119906 isin 119880 119906 lt 119908 =

(88)

By (86) 2119863(119879(119909) 119879(119910)) = 0 le 120582119869(119909 119910)

Case 2 If 119909 = 119910 = 3 then 119869(119909 119910) = 120583 and 119879(119909) =


2119863(119880119880) = 0 le 120582119869(119909 119910)



inf119908isin119882

119869(119906 119908) =


119869(119906 119908) =


inf119906isin119880


119908

cup 119880119908and

inf119906isin119880

119869 (119906 119908)

le

inf119906isin119880119908119902 (119906 119908) = 0 if 119880

119906

= 119906 isin 119880 119906 ge 119908 =

inf119906isin119880119908

(119906 minus 119908)2= 0 if 119880

119906= 119906 isin 119880 119906 lt 119908 =

(89)

Thus 2119863(119879(119909) 119879(119910)) = 0 ⩽ 120582119869(119909 119910)


(3 6]) 119910 = 119882 sub 119864 and sup119906isin119880

inf119908isin119882

119869(119906 119908) = 0 sincefor 119906 isin 119880

inf119908isin119882

119869 (119906 119908)

le

inf119908isin119882119906119902 (119906 119908) = 0 if 119882

119906

= 119908 isin 119882 119906 ge 119908 =

inf119908isin119882119906

(119906 minus 119908)2= 0 if 119882

119906= 119908 isin 119882 119906 lt 119908 =

(90)


inf119906isin119880

119869(119906 119908) = 0 since inf119906isin119880

119869(119906 119908) = 119869(119908119908) = 0 for 119908 isin 119882 Thus 2119863(119879(119909) 119879(119910)) = 0 le 120582119869(119909 119910)



exist119910isin119879(119909)

119869(119909




inf119908isin119882

119869(1199090 119908) = 0 since

inf119908isin119882

119869 (1199090 119908)

le

inf119908isin1198821199090119902 (1199090 119908) = 0 if 119882

1199090 = 119908 isin 119882 1199090 ge 119908 =

inf119908isin1198821199090

(1199090 minus 119908)2= 0 if 119882

1199090= 119908 isin 119882 1199090 lt 119908 =

(91)



12)

then 119869(1199090 1199100) lt 119869(1199090 119879(1199090)) + 120574



119869(1199090 1199100) lt 119869(1199090 119879(1199090)) + 120574




119869(1199090 119879(1199090)) + 120574


1199080


119898



sup119899gt119898

119869(119908119898

119908119899


lim119898rarrinfin

119869(119908 119908119898



1isin 119879(119908

0) = ([0 3) cup

(3 6]) 1199080 and forall

119898ge2 119908119898

isin 119879(119908119898minus1


(119908119898119898isin0cupN)

Case 2 If 1199080= 3 then 119908

1isin 119879(119908

0) = [0 3) cup (3 6] 1199082

isin

119879(1199081) = ([0 3) cup (3 6]) 119908

1 and forall

119898⩾3 119908119898

isin 119879(119908119898minus1

) sub




0isin 119883 and



119898rarrinfinsup

119899gt119898119869(119908

119898

119908119899

) =

0 of (119908119898


[2]) is a left P


119898 119898 isin N) sub 119879

[2](119883) =



119898 isin N) and (119906119898

119898 isin N)


119898isin 119879(119906


exist1198980isinN

forall119898ge1198980

119909119898


(119909119898 119898isinN)and 6 isin 119879

[2](6)

(7) ForP21 = 119901J



that 6 isin 119879[2]


0isin 119883






11 = 119901


119879 (119906) =

4 for 119906 isin (0 3)

2 for 119906 isin [3 6) (92)



119909119910isin119883119863

119871minusP111119883 (119879(119909) 119879(119910)) =

max119901(119879(119909) 119879(119910)) 119901(119879(119910) 119879(119909)) le 120582119901(119909 119910) and Fix(119879) =






119898+1= 119879(119908

119898


sup119899gt119898

119901(119908119898

119908119899

) =


119898



(119908119898119898isin0cupN)




119898 isin N) sub

119879[2]



119898 isin N) and (119906119898

119898 isin N)


119898= 119879

[2](119906

119898) then by (77) (78) and (92)



(2) isin

LIM119871minusP11(119909119898 119898isinN)

and 4 = 119879[2]





119883119860 then 119901(119906 119908) = 119901(119908 119906) = 120574 gt 0(6) For P

11 = 119901 (119883 119879) and J11 = P





119889


119889



(119883 119889) rarr (CB(119883)119867119889











References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










such that


120572(119908

0 119879 (119908

0)) lt(1minus

120582120572

119862120572

) 119903120572 (29)

Put


(0)120572

=(1minus120582120572

119862120572

) 119903120572minus 119869

120572(119908

0 119879 (119908

0)) (30)


= 120573(0)120572

120572isinA isin


1isin 119879(119908

0) such that


120572(119908

0 119908

1) lt 119869

120572(119908

0 119879 (119908

0)) + 120573

(0)120572

(31)



120572(119908

0 119908

1) lt(1minus

120582120572

119862120572

) 119903120572 (32)

Put now


(1)120572

=(120582120572

119862120572

)[(1minus120582120572

119862120572

) 119903120572minus 119869

120572(119908

0 119908

1)] (33)


(1)120572



2isin 119879(119908

1) such that


120572(119908

1 119908

2) lt 119869

120572(119908

1 119879 (119908

1)) + 120573

(1)120572

(34)

Observe that


120572(119908

1 119908

2) lt(

120582120572

119862120572

)(1minus120582120572

119862120572

) 119903120572 (35)



120572(119908

1 119908

2) lt 119869

120572(119908

1 119879 (119908

1)) + 120573

(1)120572

le sup 119869120572(119906 119879 (119908

1)) 119906 isin119879 (119908

0) + 120573

(1)120572

le119863119871minusJ119862A

1205782119883120572 (119879 (1199080) 119879 (119908

1)) + 120573

(1)120572

le(120582120572

119862120572

) 119869120572(119908

0 119908

1) + 120573

(1)120572

=(120582120572

119862120572

)(1minus120582120572

119862120572

) 119903120572

(36)



(2)120572

=(120582120572

119862120572

)[(120582120572

119862120572

)(1minus120582120572

119862120572

) 119903120572minus 119869

120572(119908

1 119908

2)]

(37)

Then in view of (35) 120573(2)= 120573

(2)120572



3isin

119879(1199082) such that


120572(119908

2 119908

3) lt 119869

120572(119908

2 119879 (119908

2)) + 120573

(2)120572

(38)



120572(119908

2 119908

3) lt(

120582120572

119862120572

)

2

(1minus120582120572

119862120572

) 119903120572 (39)



120572(119908

2 119908

3) lt 119869

120572(119908

2 119879 (119908

2)) + 120573

(2)120572

⩽ sup119906isin119879(119908

1)

119869120572(119906 119879 (119908

2)) + 120573

(2)120572

le119863119871minusJ119862A

1205782119883120572 (119879 (1199081) 119879 (119908

2)) + 120573

(2)120572

le(120582120572

119862120572

) 119869120572(119908

1 119908

2) + 120573

(2)120572

=(120582120572

119862120572

)

2

(1minus120582120572

119862120572

) 119903120572

(40)





119898+1isin119879 (119908

119898

) (41)


(119898)

=

120573(119898)

120572120572isinA

where


119898isinN 120573(119898)

120572=(

120582120572

119862120572

)

sdot [(120582120572

119862120572

)

119898minus1(1minus

120582120572

119862120572

) 119903120572minus 119869

120572(119908

119898minus1 119908

119898

)]

(42)


(119898)

isin (0infin)A


119898isinN 119869120572(119908

119898

119908119898+1

) lt 119869120572(119908

119898

119879 (119908119898

))

+ 120573(119898)

120572

(43)


119898isinN 119869120572(119908

119898

119908119898+1

)

lt(120582120572

119862120572

)

119898

(1minus120582120572

119862120572

) 119903120572

(44)



forall120572isinA

119869120572(119908

119898

119908119899

) ⩽119862120572119869120572(119908

119898

119908119898+1

)

+1198622120572119869120572(119908

119898+1 119908


+ 119862119899minus119898minus1120572

119869120572(119908

119899minus2 119908

119899minus1) +119862

119899minus119898minus1120572

119869120572(119908

119899minus1 119908

119899

)

=


119895=0119862119895+1120572

119869120572(119908

119898+119895

119908119898+119895+1

)

+119862119899minus119898minus1120572

119869120572(119908

119899minus1 119908

119899

)

(45)


119869120572(119908

119898

119908119899

) lt (1minus120582120572

119862120572

)

sdot 119903120572

[

[


119895=0119862119895+1120572

(120582120572

119862120572

)

119898+119895

+119862119899minus119898minus1120572

(120582120572

119862120572

)

119899minus2]

]

= (1minus120582120572

119862120572

)

sdot 119903120572

[

[

119862120572(

120582120572

119862120572

)

119898 119899minus119898minus2sum

119895=0120582119895

120572+(

119862120572

1205822120572

)120582119899

120572

119862119898

120572

]

]

(46)



119898isin0cupN 119908119898+1

isin119879 (119908119898

) (47)


120572le 120582

119898

120572


119898rarrinfin

sup119899gt119898

119869120572(119908

119898

119908119899


sup119899gt119898

(1minus120582120572

119862120572

)

sdot 119903120572[119862

120572(

120582120572

119862120572

)

119898

(1 minus 120582120572)minus1

+(119862120572

1205822120572

)120582119899

120572

119862119898

120572

]


(1minus120582120572

119862120572

) 119903120572[119862

120572(

120582120572

119862120572

)

119898

(1minus120582120572)minus1

+(119862120572

1205822120572

)(120582120572

119862120572

)

119898

]= 0

(48)


0isin 119883 by



119898rarrinfin

119869120572(119908 119908

119898

) = 0 (49)


= 119908119898 and 119908

119898= 119908 for 119898 isin N by



119898 isin N) and (119908119898



119898rarrinfin

119901120572(119908 119908

119898


119901120572(119908

119898 119906

119898) = 0 (50)



[119896]



119908(119898+1)119896

isin 119879[119896]

(119908119898119896


=

119908119898minus1+119896

119898 isin N) we see that (119909119898

119898 isin N) sub 119879[119896]

(119883)LIM119871minusP119862A


119898=

119908(119898+1)119896

119898 isin N) sub 119879[119896]

(119883) and (119906119898

= 119908119898119896

119898 isin N) sub

119879[119896]


119898isin 119879

[119896]


of (119909119898








[119896] is left P119862A-



such that 119908 isin 119879[119896]

(119908)



119862A)









119862A isin J119871(119883P119862A)


-


= 119863119871minusJ119862A120578119883120572

1198832



119906119908isin119883119863

119871minusJ119862A1119883120572 (119906 119908)

=max 119869120572(119906 119908) 119869

120572(119908 119906)


119906119908isin119883119863

119871minusJ119862A2119883120572 (119906 119908) = 119869

120572(119906 119908)

(51)





119909119910isin119883119862

120572sdot 119863

119871minusJ119862A120578119883120572

(119879 (119909) 119879 (119910))

le 120582120572119869120572(119909 119910)

(52)

(B) If J119862A isin J119877


D119877minusJ119862A119883120578


= 119863119877minusJ119862A120578119883120572

1198832

rarr



119906119908isin119883119863

119877minusJ119862A1119883120572 (119906 119908)

=max 119869120572(119906 119908) 119869

120572(119908 119906)


119906119908isin119883119863

119877minusJ119862A2119883120572 (119906 119908) = 119869

120572(119906 119908)

(53)




119909119910isin119883119862

120572sdot 119863

119877minusJ119862A120578119883120572

(119879 (119909) 119879 (119910))

le 120582120572119869120572(119909 119910)

(54)






(A) Given 1199080



119898

= 119879[119898]

(1199080)


(119908119898119898isin0cupN) = )

whenever


119898rarrinfin

sup119899gt119898

119869120572(119908

119898

119908119899

) = 0


119898rarrinfin

sup119899gt119898

119869120572(119908

119899

119908119898

) = 0)

(55)



isin 119883









[119896]

) is


(119909119898

119898 isin N) in 119879[119896]





119898 119898 isin N) and (119906


forall119898isinN V

119898= 119879

[119896]





119879[119896]

(119909) (119909 = 119879[119896]

(119909))


119862A)



120572120572isinA isin [0 1)A




119871minusJ119862A119883120578


119877minusJ119862A119883120578




(119908119898

= 119879[119898]



[119896]

) is left(right)P




119898

= 119879[119898]




120572(V 119879 (V)) = 119869

120572(119879 (V) V) = 0 (56)




) is left (right)P


Fix (119879[119896]

) = Fix (119879) = 119908 (57)


= 119879[119898]




120572(119908 119908) = 0 (58)




[119896]

(V) = 119879[2119896]

(V) 119879(V) =

119879[2119896]


0 lt 1198691205720

(V 119879 (V)) = 1198691205720

(119879[2119896]

(V) 119879[2119896](119879 (V)))

le 119863119871minusJ119862A1205781198831205720

(119879[2119896]

(V) 119879[2119896](119879 (V)))


le (1205821205720

1198621205720

)1198691205720

(119879[2119896minus1]

(V) 119879[2119896minus1](119879 (V)))

le (1205821205720

1198621205720

) sdot119863119871minusJ119862A1205781198831205720

(119879[2119896minus1]

(V) 119879[2119896minus1](119879 (V)))

le (1205821205720

1198621205720

)

2

1198691205720

(119879[2119896minus2]


le (1205821205720

1198621205720

)

2119896

1198691205720

(V 119879 (V)) lt 1198691205720

(V 119879 (V))

(59)



120572(V 119879 (V)) = 0 (60)




119879[119896]

(V) = 119879[2119896]


0 lt 1198691205720

(119879 (V) V) = 1198691205720

(119879[119896+1]

(V) 119879[2119896](V))

le

119896minus2sum

119898=1119862119898

12057201198691205720

(119879[119896+119898]

(V) 119879[119896+119898+1](V))

+119862119896minus21205720

1198691205720

(119879[2119896minus1]

(V) 119879[2119896](V))

le

119896minus2sum

119898=1119862119898

1205720sdot 119863

119871minusJ119862A1205781198831205720

(119879[119896+119898]

(V) 119879[119896+119898+1](V))

+119862119896minus21205720

sdot 119863119871minusJ119862A1205781198831205720

(119879[2119896minus1]

(V) 119879[2119896](V))

le

119896minus2sum

119898=1119862119898

1205720(

1205821205720

1198621205720

)

119896+119898

1198691205720

(V 119879 (V))

+119862119896minus21205720

(1205821205720

1198621205720

)

2119896minus1

1198691205720

(V 119879 (V)) = 0

(61)



120572(119879 (V) V) = 0 (62)






)


120572 120572 isin A is


120572(119879(V) V) gt



120572(119879(V) V) gt 0 or 119869

120572(V 119879(V)) gt 0 However by


120572(V V) = 0 In fact


120572(V V) le 119862

120572[119869

120572(V 119879(V))+119869

120572(119879(V) V)] = 0




1199011205720(119906 119908) gt 0 or 119901

1205720(119908 119906) gt 0 By applying


1198691205720(119906 119908) gt 0 or 119869

1205720(119908 119906) gt


exist1205720isinA

[1198691205720

(119906 119908) gt 0 1198691205720

(119906 119908)

= 1198691205720

(119879 (119906) 119879 (119908)) le119863119871minusJ119862A1205781198831205720

(119879 (119906) 119879 (119908))

le(1205821205720

1198621205720

)1198691205720

(119906 119908) lt 1198691205720

(119906 119908)] or [1198691205720

(119908 119906)

gt 0 1198691205720

(119908 119906) = 1198691205720

(119879 (119908) 119879 (119906))

le119863119871minusJ119862A1205781198831205720

(119879 (119908) 119879 (119906)) le(1205821205720

1198621205720

)1198691205720

(119908 119906)

lt 1198691205720

(119908 119906)]

(64)




rarr [0infin)

be of the form

119901 (119906 119908)

=

0 if 119906 ge 119908 119906 119908 cap (0 6) = 119906 119908

(119908 minus 119906)4 if 119906 lt 119908 119906 119908 cap (0 6) = 119906 119908

120574 if 119906 119908 cap (0 6) = 119906 119908

(65)


space In fact





8(119908 minus V)4 = 8[119901(119906 V) + 119901(V 119908)]


(119908 minus 119906)4 and 119891(V0) = min



and V0 = (119906 + 119908)2


119906119908isin(06)119906lt119908(119908 minus 119906)4

=


119906leVle1199088[119901(119906 V) + 119901(V 119908)] =

sup119906119908isin(06)119906lt119908min


(6 minus 3)4] = 8[81 + 81] = 1296




4lt

8(V minus 119906)4= 8[119901(119906 V) + 119901(V 119908)]


1296 le 8[119901(119906 V) + 119901(V 119908)] = 8[120574 + 120574]


8120574 le 8[119901(119906 V) + 119901(V 119908)]



119901(119906119908) = 119901(119908 119906) does not hold




119898= 3





(119906119898 119898isinN)= [0 3]



119901 (119906 119908) =

0 if 119860 cap 119906 119908 = 119906 119908

120574 if 119860 cap 119906 119908 = 119906 119908

(67)









119901(119906 119906) = 0 does not hold(3)P


(119906119898 119898isinN)= LIM119877minusP11

(119906119898 119898isinN)= 119860 for





119901 (119906 119908) =

0 if 119906 ge 119908

(119908 minus 119906)3 if 119906 lt 119908

(68)






4119901(V 119908) = 4[119901(119906 V) + 119901(V 119908)]


119891(V0) = min119906⩽V⩽119908119891(V) = (119908 minus 119906)

3 where V0 = (119906 + 119908)2 is


4(119908 minus 119906)[1199082+ 119908119906 + 119906

2+ 3V2 minus 3V(119908 + 119906)]



3lt 4(V minus 119906)

3= 4119901(119906 V) = 4[119901(119906

V) + 119901(V 119908)]



119901(119906119908) = 119901(119908 119906) does not hold




(119906119898 119898isinN)= [2 6] and

LIM119877minusP41(119906119898 119898isinN)

= [1 2] for sequence (119906119898



1198832


119901 (119906 119908) =

0 if 119906 ge 119908

(119906 minus 119908)2 if 119906 lt 119908

(69)

Let

119864 = [0 3) cup (3 6] (70)


2rarr [0infin) is

of the form

119869 (119906 119908) =

119901 (119906 119908) if 119864 cap 119906 119908 = 119906 119908

120583 if 119864 cap 119906 119908 = 119906 119908

(71)


36 and 119869(6 0) = 0(2)J

21 = 119869 isin J119871(119883P21)

capJ119877(119883P21)

SeeTheorem 14




119901 (119906 119908)

=

0 if 119906 ge 119908 119906 119908 cap (0 6) = 119906 119908

(119908 minus 119906)4 if 119906 lt 119908 119906 119908 cap (0 6) = 119906 119908

120574 if 119906 119908 cap (0 6) = 119906 119908

(72)


119879 (119906) =

[1 2] if 119906 isin [0 3) cup (4 6]

(4 6) if 119906 isin [3 4] (73)


Let119864 = [0 3) cup (4 6] (74)


119869 (119906 119908) =

119901 (119906 119908) if 119864 cap 119906 119908 = 119906 119908

120574 if 119864 cap 119906 119908 = 119906 119908

(75)

(1) (119883P81) where P




(2) (119883 119879) is a (119863 = D119871minusJ8112119883 = D

119877minusJ8112119883 120582 isin [2048


119909119910isin1198838 sdot

119863(119879(119909) 119879(119910)) le 120582119869(119909 119910) where

119863 (119880119882) = maxsup119906isin119880

119869 (119906119882) sup119908isin119882

119869 (119880 119908)

119880119882 isin 2119883

(76)



119864 and sup119906isin119880

inf119908isin119880

119869(119906 119908) = sup119906isin119880

119869(119906 119906) = 119901(119906 119906) =

0 = 0 Thus 4119863(119879(119909) 119879(119910)) = 0 le 120582119869(119909 119910)



inf119908isin119882

119869(119906 119908) =

sup119906isin119880

inf119908isin119882

(119908 minus 119906)4 = sup

119906isin119880(4 minus 119906)

4= 81 and

sup119908isin119882

inf119906isin119880

119869(119906 119908) = sup119908isin119882

inf119906isin119880

(119908 minus 119906)4 =

sup119908isin119882





119869(119906 119908) = sup119906isin119880

inf119908isin119882

119901(119906 119908) =

sup119908isin119882

inf119906isin119880

119869(119906 119908) = sup119908isin119882

inf119906isin119880

119901(119906 119908) = 0Therefore 8119863(119879(119909) 119879(119910)) = 0 le 120582119869(119909 119910)




exist119910isin119879(119909)

119869(119909




120573isin(0infin)119869(1199090

1199100) lt 119869(1199090 119879(1199090)) + 120573



120573isin(0infin)119869(1199090 1199100) lt 119869(1199090 119879(1199090)) + 120573


120573isin(0infin)119869(1199090 1199100) lt

119869(1199090 119879(1199090)) + 120573



119869(1199090 1199100) lt 119869(1199090 119879(1199090)) +

120573


120573isin(0infin)119869(1199090 1199100) lt 119869(1199090 119879(1199090)) + 120573


1199080isin 119883 In fact if1199080

isin 119883 and (119908119898


119898+1isin 119879(119908

119898


sup119899gt119898

119869(119908119898

119908119899

) =


sup119899gt119898

119869(119908119899

119908119898

) = 0) then forall119898ge2 119908

119898



119901(119908119908119898

) =


119898rarrinfin119901(119908

119898



119869(119908 119908119898

) =


119898rarrinfin119869(119908

119898





(119909119898 119898isinN)= (LIM119877minusP81

(119909119898 119898isinN)= ))


119898 isin N) and (119906119898

119898 isin N)


119898isin 119879(119906

119898) Then forall

119898ge2 119909119898

isin [1 2] 2 isin





following


0isin 119883




0isin 119883





119909119910isin1198838119863(119879(119909)


119901(119906119882) sup



119909119910isin1198838119863(119879(119909) 119879(119910)) le 1205820119901(119909 119910) Letting 1199090 = 2


[1 2] = 119880 119879(1199100) = (4 6) = 119882 sup119906isin[12]119901(119906 (4 6)) =

sup119906isin[12](4 minus 119906)

4= 34 = 81 and sup

119908isin(46)119901([1 2] 119908) =




119862A)


119862A whereJ



119862A)whenJ119862A = P


119862A





119860 = 1198601 cup1198602 1198601 = (0 2] 1198602 = [4 6) (77)

Let 119901 1198832


119901 (119906 119908) =

0 if 119860 cap 119906 119908 = 119906 119908

120574 if 119860 cap 119906 119908 = 119906 119908

(78)

and letJ11 = P


119879 (119906) =

1198602 for 119906 isin (0 3)

119860 for 119906 = 3

1198601 for 119906 isin (3 6)

(79)


Section 11(2) (119883 119879) is a (D



119909119910isin119883119863

119871minusP1112119883 (119879(119909) 119879(119910)) le 120582119901(119909


and maxsup119906isin119879(119909)

119901(119906 119879(119910)) sup119908isin119879(119910)



exist119910isin119879(119909)



119901 (1199090 1199100) = 119901 (1199090 119879 (1199090))

=

0 if 1199090 isin 1198601

120574 for 1199090 isin (0 3) 1198601

(80)

Therefore 119901(1199090 1199100) lt 119901(1199090 119879(1199090)) + 120573



119901 (1199090 1199100) = 119901 (1199090 119879 (1199090))

=

0 if 1199090 isin 1198602

120574 for 1199090 isin (3 6) 1198602

(81)







sup119899gt119898

119901(119908119898

119908119899


lim119898rarrinfin

119901(119908119908119898


(79)forall119898⩾1 119908

119898

isin 119879(119908119898minus1



(119908119898119898isin0cupN)




119898 isin N) sub 119879[2]

(119883) = 119860



119898 isin N) and (119906119898


119898isin 119879(119906


exist1198980isinN

forall119898⩾1198980

119909119898



and Fix(119879[2]) = 119860

(6) For (119883P11)P11 = 119901 J

11 = P11 and



isin

119883 1199081isin 119879(119908

0) and 119908

2= 119906 isin 119879(119908


and forall119898⩾3 119908

119898

= 119906 then sequence (119908119898




1198832


119901 (119906 119908) =

0 if 119906 ge 119908

(119906 minus 119908)2 if 119906 lt 119908

(82)


119879 (119906) = ([0 3) cup (3 6]) 119906 for 119906 isin [0 6] (83)

Let

119864 = [0 3) cup (3 6] (84)


2rarr [0infin) is

of the form

119869 (119906 119908) =

119901 (119906 119908) if 119864 cap 119906 119908 = 119906 119908

120583 if 119864 cap 119906 119908 = 119906 119908

(85)


119869(0 6) = 36 and 119869(6 0) = 0(2)J

21 = 119869 isin J119871(119883P21)

capJ119877(119883P21)

SeeTheorem 14




[0 1) and

119863(119880119882) = maxsup119906isin119880

119869 (119906119882) sup119908isin119882

119869 (119880 119908)

119880119882 isin 2119883

(86)




119906

cup 119882119906and

inf119908isin119882

119869 (119906 119908)

le

inf119908isin119882119906119902 (119906 119908) = 0 if 119882

119906

= 119908 isin 119882 119906 ge 119908 =

inf119908isin119882119906

(119906 minus 119908)2= 0 if 119882

119906= 119908 isin 119882 119906 lt 119908 =

(87)


cup 119880119908and

inf119906isin119880

119869 (119906 119908)

le

inf119906isin119880119908119902 (119906 119908) = 0 if 119880

119906

= 119906 isin 119880 119906 ge 119908 =

inf119906isin119880119908

(119906 minus 119908)2= 0 if 119880

119906= 119906 isin 119880 119906 lt 119908 =

(88)

By (86) 2119863(119879(119909) 119879(119910)) = 0 le 120582119869(119909 119910)

Case 2 If 119909 = 119910 = 3 then 119869(119909 119910) = 120583 and 119879(119909) =


2119863(119880119880) = 0 le 120582119869(119909 119910)



inf119908isin119882

119869(119906 119908) =


119869(119906 119908) =


inf119906isin119880


119908

cup 119880119908and

inf119906isin119880

119869 (119906 119908)

le

inf119906isin119880119908119902 (119906 119908) = 0 if 119880

119906

= 119906 isin 119880 119906 ge 119908 =

inf119906isin119880119908

(119906 minus 119908)2= 0 if 119880

119906= 119906 isin 119880 119906 lt 119908 =

(89)

Thus 2119863(119879(119909) 119879(119910)) = 0 ⩽ 120582119869(119909 119910)


(3 6]) 119910 = 119882 sub 119864 and sup119906isin119880

inf119908isin119882

119869(119906 119908) = 0 sincefor 119906 isin 119880

inf119908isin119882

119869 (119906 119908)

le

inf119908isin119882119906119902 (119906 119908) = 0 if 119882

119906

= 119908 isin 119882 119906 ge 119908 =

inf119908isin119882119906

(119906 minus 119908)2= 0 if 119882

119906= 119908 isin 119882 119906 lt 119908 =

(90)


inf119906isin119880

119869(119906 119908) = 0 since inf119906isin119880

119869(119906 119908) = 119869(119908119908) = 0 for 119908 isin 119882 Thus 2119863(119879(119909) 119879(119910)) = 0 le 120582119869(119909 119910)



exist119910isin119879(119909)

119869(119909




inf119908isin119882

119869(1199090 119908) = 0 since

inf119908isin119882

119869 (1199090 119908)

le

inf119908isin1198821199090119902 (1199090 119908) = 0 if 119882

1199090 = 119908 isin 119882 1199090 ge 119908 =

inf119908isin1198821199090

(1199090 minus 119908)2= 0 if 119882

1199090= 119908 isin 119882 1199090 lt 119908 =

(91)



12)

then 119869(1199090 1199100) lt 119869(1199090 119879(1199090)) + 120574



119869(1199090 1199100) lt 119869(1199090 119879(1199090)) + 120574




119869(1199090 119879(1199090)) + 120574


1199080


119898



sup119899gt119898

119869(119908119898

119908119899


lim119898rarrinfin

119869(119908 119908119898



1isin 119879(119908

0) = ([0 3) cup

(3 6]) 1199080 and forall

119898ge2 119908119898

isin 119879(119908119898minus1


(119908119898119898isin0cupN)

Case 2 If 1199080= 3 then 119908

1isin 119879(119908

0) = [0 3) cup (3 6] 1199082

isin

119879(1199081) = ([0 3) cup (3 6]) 119908

1 and forall

119898⩾3 119908119898

isin 119879(119908119898minus1

) sub




0isin 119883 and



119898rarrinfinsup

119899gt119898119869(119908

119898

119908119899

) =

0 of (119908119898


[2]) is a left P


119898 119898 isin N) sub 119879

[2](119883) =



119898 isin N) and (119906119898

119898 isin N)


119898isin 119879(119906


exist1198980isinN

forall119898ge1198980

119909119898


(119909119898 119898isinN)and 6 isin 119879

[2](6)

(7) ForP21 = 119901J



that 6 isin 119879[2]


0isin 119883






11 = 119901


119879 (119906) =

4 for 119906 isin (0 3)

2 for 119906 isin [3 6) (92)



119909119910isin119883119863

119871minusP111119883 (119879(119909) 119879(119910)) =

max119901(119879(119909) 119879(119910)) 119901(119879(119910) 119879(119909)) le 120582119901(119909 119910) and Fix(119879) =






119898+1= 119879(119908

119898


sup119899gt119898

119901(119908119898

119908119899

) =


119898



(119908119898119898isin0cupN)




119898 isin N) sub

119879[2]



119898 isin N) and (119906119898

119898 isin N)


119898= 119879

[2](119906

119898) then by (77) (78) and (92)



(2) isin

LIM119871minusP11(119909119898 119898isinN)

and 4 = 119879[2]





119883119860 then 119901(119906 119908) = 119901(119908 119906) = 120574 gt 0(6) For P

11 = 119901 (119883 119879) and J11 = P





119889


119889



(119883 119889) rarr (CB(119883)119867119889











References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











forall120572isinA

119869120572(119908

119898

119908119899

) ⩽119862120572119869120572(119908

119898

119908119898+1

)

+1198622120572119869120572(119908

119898+1 119908


+ 119862119899minus119898minus1120572

119869120572(119908

119899minus2 119908

119899minus1) +119862

119899minus119898minus1120572

119869120572(119908

119899minus1 119908

119899

)

=


119895=0119862119895+1120572

119869120572(119908

119898+119895

119908119898+119895+1

)

+119862119899minus119898minus1120572

119869120572(119908

119899minus1 119908

119899

)

(45)


119869120572(119908

119898

119908119899

) lt (1minus120582120572

119862120572

)

sdot 119903120572

[

[


119895=0119862119895+1120572

(120582120572

119862120572

)

119898+119895

+119862119899minus119898minus1120572

(120582120572

119862120572

)

119899minus2]

]

= (1minus120582120572

119862120572

)

sdot 119903120572

[

[

119862120572(

120582120572

119862120572

)

119898 119899minus119898minus2sum

119895=0120582119895

120572+(

119862120572

1205822120572

)120582119899

120572

119862119898

120572

]

]

(46)



119898isin0cupN 119908119898+1

isin119879 (119908119898

) (47)


120572le 120582

119898

120572


119898rarrinfin

sup119899gt119898

119869120572(119908

119898

119908119899


sup119899gt119898

(1minus120582120572

119862120572

)

sdot 119903120572[119862

120572(

120582120572

119862120572

)

119898

(1 minus 120582120572)minus1

+(119862120572

1205822120572

)120582119899

120572

119862119898

120572

]


(1minus120582120572

119862120572

) 119903120572[119862

120572(

120582120572

119862120572

)

119898

(1minus120582120572)minus1

+(119862120572

1205822120572

)(120582120572

119862120572

)

119898

]= 0

(48)


0isin 119883 by



119898rarrinfin

119869120572(119908 119908

119898

) = 0 (49)


= 119908119898 and 119908

119898= 119908 for 119898 isin N by



119898 isin N) and (119908119898



119898rarrinfin

119901120572(119908 119908

119898


119901120572(119908

119898 119906

119898) = 0 (50)



[119896]



119908(119898+1)119896

isin 119879[119896]

(119908119898119896


=

119908119898minus1+119896

119898 isin N) we see that (119909119898

119898 isin N) sub 119879[119896]

(119883)LIM119871minusP119862A


119898=

119908(119898+1)119896

119898 isin N) sub 119879[119896]

(119883) and (119906119898

= 119908119898119896

119898 isin N) sub

119879[119896]


119898isin 119879

[119896]


of (119909119898








[119896] is left P119862A-



such that 119908 isin 119879[119896]

(119908)



119862A)









119862A isin J119871(119883P119862A)


-


= 119863119871minusJ119862A120578119883120572

1198832



119906119908isin119883119863

119871minusJ119862A1119883120572 (119906 119908)

=max 119869120572(119906 119908) 119869

120572(119908 119906)


119906119908isin119883119863

119871minusJ119862A2119883120572 (119906 119908) = 119869

120572(119906 119908)

(51)





119909119910isin119883119862

120572sdot 119863

119871minusJ119862A120578119883120572

(119879 (119909) 119879 (119910))

le 120582120572119869120572(119909 119910)

(52)

(B) If J119862A isin J119877


D119877minusJ119862A119883120578


= 119863119877minusJ119862A120578119883120572

1198832

rarr



119906119908isin119883119863

119877minusJ119862A1119883120572 (119906 119908)

=max 119869120572(119906 119908) 119869

120572(119908 119906)


119906119908isin119883119863

119877minusJ119862A2119883120572 (119906 119908) = 119869

120572(119906 119908)

(53)




119909119910isin119883119862

120572sdot 119863

119877minusJ119862A120578119883120572

(119879 (119909) 119879 (119910))

le 120582120572119869120572(119909 119910)

(54)






(A) Given 1199080



119898

= 119879[119898]

(1199080)


(119908119898119898isin0cupN) = )

whenever


119898rarrinfin

sup119899gt119898

119869120572(119908

119898

119908119899

) = 0


119898rarrinfin

sup119899gt119898

119869120572(119908

119899

119908119898

) = 0)

(55)



isin 119883









[119896]

) is


(119909119898

119898 isin N) in 119879[119896]





119898 119898 isin N) and (119906


forall119898isinN V

119898= 119879

[119896]





119879[119896]

(119909) (119909 = 119879[119896]

(119909))


119862A)



120572120572isinA isin [0 1)A




119871minusJ119862A119883120578


119877minusJ119862A119883120578




(119908119898

= 119879[119898]



[119896]

) is left(right)P




119898

= 119879[119898]




120572(V 119879 (V)) = 119869

120572(119879 (V) V) = 0 (56)




) is left (right)P


Fix (119879[119896]

) = Fix (119879) = 119908 (57)


= 119879[119898]




120572(119908 119908) = 0 (58)




[119896]

(V) = 119879[2119896]

(V) 119879(V) =

119879[2119896]


0 lt 1198691205720

(V 119879 (V)) = 1198691205720

(119879[2119896]

(V) 119879[2119896](119879 (V)))

le 119863119871minusJ119862A1205781198831205720

(119879[2119896]

(V) 119879[2119896](119879 (V)))


le (1205821205720

1198621205720

)1198691205720

(119879[2119896minus1]

(V) 119879[2119896minus1](119879 (V)))

le (1205821205720

1198621205720

) sdot119863119871minusJ119862A1205781198831205720

(119879[2119896minus1]

(V) 119879[2119896minus1](119879 (V)))

le (1205821205720

1198621205720

)

2

1198691205720

(119879[2119896minus2]


le (1205821205720

1198621205720

)

2119896

1198691205720

(V 119879 (V)) lt 1198691205720

(V 119879 (V))

(59)



120572(V 119879 (V)) = 0 (60)




119879[119896]

(V) = 119879[2119896]


0 lt 1198691205720

(119879 (V) V) = 1198691205720

(119879[119896+1]

(V) 119879[2119896](V))

le

119896minus2sum

119898=1119862119898

12057201198691205720

(119879[119896+119898]

(V) 119879[119896+119898+1](V))

+119862119896minus21205720

1198691205720

(119879[2119896minus1]

(V) 119879[2119896](V))

le

119896minus2sum

119898=1119862119898

1205720sdot 119863

119871minusJ119862A1205781198831205720

(119879[119896+119898]

(V) 119879[119896+119898+1](V))

+119862119896minus21205720

sdot 119863119871minusJ119862A1205781198831205720

(119879[2119896minus1]

(V) 119879[2119896](V))

le

119896minus2sum

119898=1119862119898

1205720(

1205821205720

1198621205720

)

119896+119898

1198691205720

(V 119879 (V))

+119862119896minus21205720

(1205821205720

1198621205720

)

2119896minus1

1198691205720

(V 119879 (V)) = 0

(61)



120572(119879 (V) V) = 0 (62)






)


120572 120572 isin A is


120572(119879(V) V) gt



120572(119879(V) V) gt 0 or 119869

120572(V 119879(V)) gt 0 However by


120572(V V) = 0 In fact


120572(V V) le 119862

120572[119869

120572(V 119879(V))+119869

120572(119879(V) V)] = 0




1199011205720(119906 119908) gt 0 or 119901

1205720(119908 119906) gt 0 By applying


1198691205720(119906 119908) gt 0 or 119869

1205720(119908 119906) gt


exist1205720isinA

[1198691205720

(119906 119908) gt 0 1198691205720

(119906 119908)

= 1198691205720

(119879 (119906) 119879 (119908)) le119863119871minusJ119862A1205781198831205720

(119879 (119906) 119879 (119908))

le(1205821205720

1198621205720

)1198691205720

(119906 119908) lt 1198691205720

(119906 119908)] or [1198691205720

(119908 119906)

gt 0 1198691205720

(119908 119906) = 1198691205720

(119879 (119908) 119879 (119906))

le119863119871minusJ119862A1205781198831205720

(119879 (119908) 119879 (119906)) le(1205821205720

1198621205720

)1198691205720

(119908 119906)

lt 1198691205720

(119908 119906)]

(64)




rarr [0infin)

be of the form

119901 (119906 119908)

=

0 if 119906 ge 119908 119906 119908 cap (0 6) = 119906 119908

(119908 minus 119906)4 if 119906 lt 119908 119906 119908 cap (0 6) = 119906 119908

120574 if 119906 119908 cap (0 6) = 119906 119908

(65)


space In fact





8(119908 minus V)4 = 8[119901(119906 V) + 119901(V 119908)]


(119908 minus 119906)4 and 119891(V0) = min



and V0 = (119906 + 119908)2


119906119908isin(06)119906lt119908(119908 minus 119906)4

=


119906leVle1199088[119901(119906 V) + 119901(V 119908)] =

sup119906119908isin(06)119906lt119908min


(6 minus 3)4] = 8[81 + 81] = 1296




4lt

8(V minus 119906)4= 8[119901(119906 V) + 119901(V 119908)]


1296 le 8[119901(119906 V) + 119901(V 119908)] = 8[120574 + 120574]


8120574 le 8[119901(119906 V) + 119901(V 119908)]



119901(119906119908) = 119901(119908 119906) does not hold




119898= 3





(119906119898 119898isinN)= [0 3]



119901 (119906 119908) =

0 if 119860 cap 119906 119908 = 119906 119908

120574 if 119860 cap 119906 119908 = 119906 119908

(67)









119901(119906 119906) = 0 does not hold(3)P


(119906119898 119898isinN)= LIM119877minusP11

(119906119898 119898isinN)= 119860 for





119901 (119906 119908) =

0 if 119906 ge 119908

(119908 minus 119906)3 if 119906 lt 119908

(68)






4119901(V 119908) = 4[119901(119906 V) + 119901(V 119908)]


119891(V0) = min119906⩽V⩽119908119891(V) = (119908 minus 119906)

3 where V0 = (119906 + 119908)2 is


4(119908 minus 119906)[1199082+ 119908119906 + 119906

2+ 3V2 minus 3V(119908 + 119906)]



3lt 4(V minus 119906)

3= 4119901(119906 V) = 4[119901(119906

V) + 119901(V 119908)]



119901(119906119908) = 119901(119908 119906) does not hold




(119906119898 119898isinN)= [2 6] and

LIM119877minusP41(119906119898 119898isinN)

= [1 2] for sequence (119906119898



1198832


119901 (119906 119908) =

0 if 119906 ge 119908

(119906 minus 119908)2 if 119906 lt 119908

(69)

Let

119864 = [0 3) cup (3 6] (70)


2rarr [0infin) is

of the form

119869 (119906 119908) =

119901 (119906 119908) if 119864 cap 119906 119908 = 119906 119908

120583 if 119864 cap 119906 119908 = 119906 119908

(71)


36 and 119869(6 0) = 0(2)J

21 = 119869 isin J119871(119883P21)

capJ119877(119883P21)

SeeTheorem 14




119901 (119906 119908)

=

0 if 119906 ge 119908 119906 119908 cap (0 6) = 119906 119908

(119908 minus 119906)4 if 119906 lt 119908 119906 119908 cap (0 6) = 119906 119908

120574 if 119906 119908 cap (0 6) = 119906 119908

(72)


119879 (119906) =

[1 2] if 119906 isin [0 3) cup (4 6]

(4 6) if 119906 isin [3 4] (73)


Let119864 = [0 3) cup (4 6] (74)


119869 (119906 119908) =

119901 (119906 119908) if 119864 cap 119906 119908 = 119906 119908

120574 if 119864 cap 119906 119908 = 119906 119908

(75)

(1) (119883P81) where P




(2) (119883 119879) is a (119863 = D119871minusJ8112119883 = D

119877minusJ8112119883 120582 isin [2048


119909119910isin1198838 sdot

119863(119879(119909) 119879(119910)) le 120582119869(119909 119910) where

119863 (119880119882) = maxsup119906isin119880

119869 (119906119882) sup119908isin119882

119869 (119880 119908)

119880119882 isin 2119883

(76)



119864 and sup119906isin119880

inf119908isin119880

119869(119906 119908) = sup119906isin119880

119869(119906 119906) = 119901(119906 119906) =

0 = 0 Thus 4119863(119879(119909) 119879(119910)) = 0 le 120582119869(119909 119910)



inf119908isin119882

119869(119906 119908) =

sup119906isin119880

inf119908isin119882

(119908 minus 119906)4 = sup

119906isin119880(4 minus 119906)

4= 81 and

sup119908isin119882

inf119906isin119880

119869(119906 119908) = sup119908isin119882

inf119906isin119880

(119908 minus 119906)4 =

sup119908isin119882





119869(119906 119908) = sup119906isin119880

inf119908isin119882

119901(119906 119908) =

sup119908isin119882

inf119906isin119880

119869(119906 119908) = sup119908isin119882

inf119906isin119880

119901(119906 119908) = 0Therefore 8119863(119879(119909) 119879(119910)) = 0 le 120582119869(119909 119910)




exist119910isin119879(119909)

119869(119909




120573isin(0infin)119869(1199090

1199100) lt 119869(1199090 119879(1199090)) + 120573



120573isin(0infin)119869(1199090 1199100) lt 119869(1199090 119879(1199090)) + 120573


120573isin(0infin)119869(1199090 1199100) lt

119869(1199090 119879(1199090)) + 120573



119869(1199090 1199100) lt 119869(1199090 119879(1199090)) +

120573


120573isin(0infin)119869(1199090 1199100) lt 119869(1199090 119879(1199090)) + 120573


1199080isin 119883 In fact if1199080

isin 119883 and (119908119898


119898+1isin 119879(119908

119898


sup119899gt119898

119869(119908119898

119908119899

) =


sup119899gt119898

119869(119908119899

119908119898

) = 0) then forall119898ge2 119908

119898



119901(119908119908119898

) =


119898rarrinfin119901(119908

119898



119869(119908 119908119898

) =


119898rarrinfin119869(119908

119898





(119909119898 119898isinN)= (LIM119877minusP81

(119909119898 119898isinN)= ))


119898 isin N) and (119906119898

119898 isin N)


119898isin 119879(119906

119898) Then forall

119898ge2 119909119898

isin [1 2] 2 isin





following


0isin 119883




0isin 119883





119909119910isin1198838119863(119879(119909)


119901(119906119882) sup



119909119910isin1198838119863(119879(119909) 119879(119910)) le 1205820119901(119909 119910) Letting 1199090 = 2


[1 2] = 119880 119879(1199100) = (4 6) = 119882 sup119906isin[12]119901(119906 (4 6)) =

sup119906isin[12](4 minus 119906)

4= 34 = 81 and sup

119908isin(46)119901([1 2] 119908) =




119862A)


119862A whereJ



119862A)whenJ119862A = P


119862A





119860 = 1198601 cup1198602 1198601 = (0 2] 1198602 = [4 6) (77)

Let 119901 1198832


119901 (119906 119908) =

0 if 119860 cap 119906 119908 = 119906 119908

120574 if 119860 cap 119906 119908 = 119906 119908

(78)

and letJ11 = P


119879 (119906) =

1198602 for 119906 isin (0 3)

119860 for 119906 = 3

1198601 for 119906 isin (3 6)

(79)


Section 11(2) (119883 119879) is a (D



119909119910isin119883119863

119871minusP1112119883 (119879(119909) 119879(119910)) le 120582119901(119909


and maxsup119906isin119879(119909)

119901(119906 119879(119910)) sup119908isin119879(119910)



exist119910isin119879(119909)



119901 (1199090 1199100) = 119901 (1199090 119879 (1199090))

=

0 if 1199090 isin 1198601

120574 for 1199090 isin (0 3) 1198601

(80)

Therefore 119901(1199090 1199100) lt 119901(1199090 119879(1199090)) + 120573



119901 (1199090 1199100) = 119901 (1199090 119879 (1199090))

=

0 if 1199090 isin 1198602

120574 for 1199090 isin (3 6) 1198602

(81)







sup119899gt119898

119901(119908119898

119908119899


lim119898rarrinfin

119901(119908119908119898


(79)forall119898⩾1 119908

119898

isin 119879(119908119898minus1



(119908119898119898isin0cupN)




119898 isin N) sub 119879[2]

(119883) = 119860



119898 isin N) and (119906119898


119898isin 119879(119906


exist1198980isinN

forall119898⩾1198980

119909119898



and Fix(119879[2]) = 119860

(6) For (119883P11)P11 = 119901 J

11 = P11 and



isin

119883 1199081isin 119879(119908

0) and 119908

2= 119906 isin 119879(119908


and forall119898⩾3 119908

119898

= 119906 then sequence (119908119898




1198832


119901 (119906 119908) =

0 if 119906 ge 119908

(119906 minus 119908)2 if 119906 lt 119908

(82)


119879 (119906) = ([0 3) cup (3 6]) 119906 for 119906 isin [0 6] (83)

Let

119864 = [0 3) cup (3 6] (84)


2rarr [0infin) is

of the form

119869 (119906 119908) =

119901 (119906 119908) if 119864 cap 119906 119908 = 119906 119908

120583 if 119864 cap 119906 119908 = 119906 119908

(85)


119869(0 6) = 36 and 119869(6 0) = 0(2)J

21 = 119869 isin J119871(119883P21)

capJ119877(119883P21)

SeeTheorem 14




[0 1) and

119863(119880119882) = maxsup119906isin119880

119869 (119906119882) sup119908isin119882

119869 (119880 119908)

119880119882 isin 2119883

(86)




119906

cup 119882119906and

inf119908isin119882

119869 (119906 119908)

le

inf119908isin119882119906119902 (119906 119908) = 0 if 119882

119906

= 119908 isin 119882 119906 ge 119908 =

inf119908isin119882119906

(119906 minus 119908)2= 0 if 119882

119906= 119908 isin 119882 119906 lt 119908 =

(87)


cup 119880119908and

inf119906isin119880

119869 (119906 119908)

le

inf119906isin119880119908119902 (119906 119908) = 0 if 119880

119906

= 119906 isin 119880 119906 ge 119908 =

inf119906isin119880119908

(119906 minus 119908)2= 0 if 119880

119906= 119906 isin 119880 119906 lt 119908 =

(88)

By (86) 2119863(119879(119909) 119879(119910)) = 0 le 120582119869(119909 119910)

Case 2 If 119909 = 119910 = 3 then 119869(119909 119910) = 120583 and 119879(119909) =


2119863(119880119880) = 0 le 120582119869(119909 119910)



inf119908isin119882

119869(119906 119908) =


119869(119906 119908) =


inf119906isin119880


119908

cup 119880119908and

inf119906isin119880

119869 (119906 119908)

le

inf119906isin119880119908119902 (119906 119908) = 0 if 119880

119906

= 119906 isin 119880 119906 ge 119908 =

inf119906isin119880119908

(119906 minus 119908)2= 0 if 119880

119906= 119906 isin 119880 119906 lt 119908 =

(89)

Thus 2119863(119879(119909) 119879(119910)) = 0 ⩽ 120582119869(119909 119910)


(3 6]) 119910 = 119882 sub 119864 and sup119906isin119880

inf119908isin119882

119869(119906 119908) = 0 sincefor 119906 isin 119880

inf119908isin119882

119869 (119906 119908)

le

inf119908isin119882119906119902 (119906 119908) = 0 if 119882

119906

= 119908 isin 119882 119906 ge 119908 =

inf119908isin119882119906

(119906 minus 119908)2= 0 if 119882

119906= 119908 isin 119882 119906 lt 119908 =

(90)


inf119906isin119880

119869(119906 119908) = 0 since inf119906isin119880

119869(119906 119908) = 119869(119908119908) = 0 for 119908 isin 119882 Thus 2119863(119879(119909) 119879(119910)) = 0 le 120582119869(119909 119910)



exist119910isin119879(119909)

119869(119909




inf119908isin119882

119869(1199090 119908) = 0 since

inf119908isin119882

119869 (1199090 119908)

le

inf119908isin1198821199090119902 (1199090 119908) = 0 if 119882

1199090 = 119908 isin 119882 1199090 ge 119908 =

inf119908isin1198821199090

(1199090 minus 119908)2= 0 if 119882

1199090= 119908 isin 119882 1199090 lt 119908 =

(91)



12)

then 119869(1199090 1199100) lt 119869(1199090 119879(1199090)) + 120574



119869(1199090 1199100) lt 119869(1199090 119879(1199090)) + 120574




119869(1199090 119879(1199090)) + 120574


1199080


119898



sup119899gt119898

119869(119908119898

119908119899


lim119898rarrinfin

119869(119908 119908119898



1isin 119879(119908

0) = ([0 3) cup

(3 6]) 1199080 and forall

119898ge2 119908119898

isin 119879(119908119898minus1


(119908119898119898isin0cupN)

Case 2 If 1199080= 3 then 119908

1isin 119879(119908

0) = [0 3) cup (3 6] 1199082

isin

119879(1199081) = ([0 3) cup (3 6]) 119908

1 and forall

119898⩾3 119908119898

isin 119879(119908119898minus1

) sub




0isin 119883 and



119898rarrinfinsup

119899gt119898119869(119908

119898

119908119899

) =

0 of (119908119898


[2]) is a left P


119898 119898 isin N) sub 119879

[2](119883) =



119898 isin N) and (119906119898

119898 isin N)


119898isin 119879(119906


exist1198980isinN

forall119898ge1198980

119909119898


(119909119898 119898isinN)and 6 isin 119879

[2](6)

(7) ForP21 = 119901J



that 6 isin 119879[2]


0isin 119883






11 = 119901


119879 (119906) =

4 for 119906 isin (0 3)

2 for 119906 isin [3 6) (92)



119909119910isin119883119863

119871minusP111119883 (119879(119909) 119879(119910)) =

max119901(119879(119909) 119879(119910)) 119901(119879(119910) 119879(119909)) le 120582119901(119909 119910) and Fix(119879) =






119898+1= 119879(119908

119898


sup119899gt119898

119901(119908119898

119908119899

) =


119898



(119908119898119898isin0cupN)




119898 isin N) sub

119879[2]



119898 isin N) and (119906119898

119898 isin N)


119898= 119879

[2](119906

119898) then by (77) (78) and (92)



(2) isin

LIM119871minusP11(119909119898 119898isinN)

and 4 = 119879[2]





119883119860 then 119901(119906 119908) = 119901(119908 119906) = 120574 gt 0(6) For P

11 = 119901 (119883 119879) and J11 = P





119889


119889



(119883 119889) rarr (CB(119883)119867119889











References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













119909119910isin119883119862

120572sdot 119863

119871minusJ119862A120578119883120572

(119879 (119909) 119879 (119910))

le 120582120572119869120572(119909 119910)

(52)

(B) If J119862A isin J119877


D119877minusJ119862A119883120578


= 119863119877minusJ119862A120578119883120572

1198832

rarr



119906119908isin119883119863

119877minusJ119862A1119883120572 (119906 119908)

=max 119869120572(119906 119908) 119869

120572(119908 119906)


119906119908isin119883119863

119877minusJ119862A2119883120572 (119906 119908) = 119869

120572(119906 119908)

(53)




119909119910isin119883119862

120572sdot 119863

119877minusJ119862A120578119883120572

(119879 (119909) 119879 (119910))

le 120582120572119869120572(119909 119910)

(54)






(A) Given 1199080



119898

= 119879[119898]

(1199080)


(119908119898119898isin0cupN) = )

whenever


119898rarrinfin

sup119899gt119898

119869120572(119908

119898

119908119899

) = 0


119898rarrinfin

sup119899gt119898

119869120572(119908

119899

119908119898

) = 0)

(55)



isin 119883









[119896]

) is


(119909119898

119898 isin N) in 119879[119896]





119898 119898 isin N) and (119906


forall119898isinN V

119898= 119879

[119896]





119879[119896]

(119909) (119909 = 119879[119896]

(119909))


119862A)



120572120572isinA isin [0 1)A




119871minusJ119862A119883120578


119877minusJ119862A119883120578




(119908119898

= 119879[119898]



[119896]

) is left(right)P




119898

= 119879[119898]




120572(V 119879 (V)) = 119869

120572(119879 (V) V) = 0 (56)




) is left (right)P


Fix (119879[119896]

) = Fix (119879) = 119908 (57)


= 119879[119898]




120572(119908 119908) = 0 (58)




[119896]

(V) = 119879[2119896]

(V) 119879(V) =

119879[2119896]


0 lt 1198691205720

(V 119879 (V)) = 1198691205720

(119879[2119896]

(V) 119879[2119896](119879 (V)))

le 119863119871minusJ119862A1205781198831205720

(119879[2119896]

(V) 119879[2119896](119879 (V)))


le (1205821205720

1198621205720

)1198691205720

(119879[2119896minus1]

(V) 119879[2119896minus1](119879 (V)))

le (1205821205720

1198621205720

) sdot119863119871minusJ119862A1205781198831205720

(119879[2119896minus1]

(V) 119879[2119896minus1](119879 (V)))

le (1205821205720

1198621205720

)

2

1198691205720

(119879[2119896minus2]


le (1205821205720

1198621205720

)

2119896

1198691205720

(V 119879 (V)) lt 1198691205720

(V 119879 (V))

(59)



120572(V 119879 (V)) = 0 (60)




119879[119896]

(V) = 119879[2119896]


0 lt 1198691205720

(119879 (V) V) = 1198691205720

(119879[119896+1]

(V) 119879[2119896](V))

le

119896minus2sum

119898=1119862119898

12057201198691205720

(119879[119896+119898]

(V) 119879[119896+119898+1](V))

+119862119896minus21205720

1198691205720

(119879[2119896minus1]

(V) 119879[2119896](V))

le

119896minus2sum

119898=1119862119898

1205720sdot 119863

119871minusJ119862A1205781198831205720

(119879[119896+119898]

(V) 119879[119896+119898+1](V))

+119862119896minus21205720

sdot 119863119871minusJ119862A1205781198831205720

(119879[2119896minus1]

(V) 119879[2119896](V))

le

119896minus2sum

119898=1119862119898

1205720(

1205821205720

1198621205720

)

119896+119898

1198691205720

(V 119879 (V))

+119862119896minus21205720

(1205821205720

1198621205720

)

2119896minus1

1198691205720

(V 119879 (V)) = 0

(61)



120572(119879 (V) V) = 0 (62)






)


120572 120572 isin A is


120572(119879(V) V) gt



120572(119879(V) V) gt 0 or 119869

120572(V 119879(V)) gt 0 However by


120572(V V) = 0 In fact


120572(V V) le 119862

120572[119869

120572(V 119879(V))+119869

120572(119879(V) V)] = 0




1199011205720(119906 119908) gt 0 or 119901

1205720(119908 119906) gt 0 By applying


1198691205720(119906 119908) gt 0 or 119869

1205720(119908 119906) gt


exist1205720isinA

[1198691205720

(119906 119908) gt 0 1198691205720

(119906 119908)

= 1198691205720

(119879 (119906) 119879 (119908)) le119863119871minusJ119862A1205781198831205720

(119879 (119906) 119879 (119908))

le(1205821205720

1198621205720

)1198691205720

(119906 119908) lt 1198691205720

(119906 119908)] or [1198691205720

(119908 119906)

gt 0 1198691205720

(119908 119906) = 1198691205720

(119879 (119908) 119879 (119906))

le119863119871minusJ119862A1205781198831205720

(119879 (119908) 119879 (119906)) le(1205821205720

1198621205720

)1198691205720

(119908 119906)

lt 1198691205720

(119908 119906)]

(64)




rarr [0infin)

be of the form

119901 (119906 119908)

=

0 if 119906 ge 119908 119906 119908 cap (0 6) = 119906 119908

(119908 minus 119906)4 if 119906 lt 119908 119906 119908 cap (0 6) = 119906 119908

120574 if 119906 119908 cap (0 6) = 119906 119908

(65)


space In fact





8(119908 minus V)4 = 8[119901(119906 V) + 119901(V 119908)]


(119908 minus 119906)4 and 119891(V0) = min



and V0 = (119906 + 119908)2


119906119908isin(06)119906lt119908(119908 minus 119906)4

=


119906leVle1199088[119901(119906 V) + 119901(V 119908)] =

sup119906119908isin(06)119906lt119908min


(6 minus 3)4] = 8[81 + 81] = 1296




4lt

8(V minus 119906)4= 8[119901(119906 V) + 119901(V 119908)]


1296 le 8[119901(119906 V) + 119901(V 119908)] = 8[120574 + 120574]


8120574 le 8[119901(119906 V) + 119901(V 119908)]



119901(119906119908) = 119901(119908 119906) does not hold




119898= 3





(119906119898 119898isinN)= [0 3]



119901 (119906 119908) =

0 if 119860 cap 119906 119908 = 119906 119908

120574 if 119860 cap 119906 119908 = 119906 119908

(67)









119901(119906 119906) = 0 does not hold(3)P


(119906119898 119898isinN)= LIM119877minusP11

(119906119898 119898isinN)= 119860 for





119901 (119906 119908) =

0 if 119906 ge 119908

(119908 minus 119906)3 if 119906 lt 119908

(68)






4119901(V 119908) = 4[119901(119906 V) + 119901(V 119908)]


119891(V0) = min119906⩽V⩽119908119891(V) = (119908 minus 119906)

3 where V0 = (119906 + 119908)2 is


4(119908 minus 119906)[1199082+ 119908119906 + 119906

2+ 3V2 minus 3V(119908 + 119906)]



3lt 4(V minus 119906)

3= 4119901(119906 V) = 4[119901(119906

V) + 119901(V 119908)]



119901(119906119908) = 119901(119908 119906) does not hold




(119906119898 119898isinN)= [2 6] and

LIM119877minusP41(119906119898 119898isinN)

= [1 2] for sequence (119906119898



1198832


119901 (119906 119908) =

0 if 119906 ge 119908

(119906 minus 119908)2 if 119906 lt 119908

(69)

Let

119864 = [0 3) cup (3 6] (70)


2rarr [0infin) is

of the form

119869 (119906 119908) =

119901 (119906 119908) if 119864 cap 119906 119908 = 119906 119908

120583 if 119864 cap 119906 119908 = 119906 119908

(71)


36 and 119869(6 0) = 0(2)J

21 = 119869 isin J119871(119883P21)

capJ119877(119883P21)

SeeTheorem 14




119901 (119906 119908)

=

0 if 119906 ge 119908 119906 119908 cap (0 6) = 119906 119908

(119908 minus 119906)4 if 119906 lt 119908 119906 119908 cap (0 6) = 119906 119908

120574 if 119906 119908 cap (0 6) = 119906 119908

(72)


119879 (119906) =

[1 2] if 119906 isin [0 3) cup (4 6]

(4 6) if 119906 isin [3 4] (73)


Let119864 = [0 3) cup (4 6] (74)


119869 (119906 119908) =

119901 (119906 119908) if 119864 cap 119906 119908 = 119906 119908

120574 if 119864 cap 119906 119908 = 119906 119908

(75)

(1) (119883P81) where P




(2) (119883 119879) is a (119863 = D119871minusJ8112119883 = D

119877minusJ8112119883 120582 isin [2048


119909119910isin1198838 sdot

119863(119879(119909) 119879(119910)) le 120582119869(119909 119910) where

119863 (119880119882) = maxsup119906isin119880

119869 (119906119882) sup119908isin119882

119869 (119880 119908)

119880119882 isin 2119883

(76)



119864 and sup119906isin119880

inf119908isin119880

119869(119906 119908) = sup119906isin119880

119869(119906 119906) = 119901(119906 119906) =

0 = 0 Thus 4119863(119879(119909) 119879(119910)) = 0 le 120582119869(119909 119910)



inf119908isin119882

119869(119906 119908) =

sup119906isin119880

inf119908isin119882

(119908 minus 119906)4 = sup

119906isin119880(4 minus 119906)

4= 81 and

sup119908isin119882

inf119906isin119880

119869(119906 119908) = sup119908isin119882

inf119906isin119880

(119908 minus 119906)4 =

sup119908isin119882





119869(119906 119908) = sup119906isin119880

inf119908isin119882

119901(119906 119908) =

sup119908isin119882

inf119906isin119880

119869(119906 119908) = sup119908isin119882

inf119906isin119880

119901(119906 119908) = 0Therefore 8119863(119879(119909) 119879(119910)) = 0 le 120582119869(119909 119910)




exist119910isin119879(119909)

119869(119909




120573isin(0infin)119869(1199090

1199100) lt 119869(1199090 119879(1199090)) + 120573



120573isin(0infin)119869(1199090 1199100) lt 119869(1199090 119879(1199090)) + 120573


120573isin(0infin)119869(1199090 1199100) lt

119869(1199090 119879(1199090)) + 120573



119869(1199090 1199100) lt 119869(1199090 119879(1199090)) +

120573


120573isin(0infin)119869(1199090 1199100) lt 119869(1199090 119879(1199090)) + 120573


1199080isin 119883 In fact if1199080

isin 119883 and (119908119898


119898+1isin 119879(119908

119898


sup119899gt119898

119869(119908119898

119908119899

) =


sup119899gt119898

119869(119908119899

119908119898

) = 0) then forall119898ge2 119908

119898



119901(119908119908119898

) =


119898rarrinfin119901(119908

119898



119869(119908 119908119898

) =


119898rarrinfin119869(119908

119898





(119909119898 119898isinN)= (LIM119877minusP81

(119909119898 119898isinN)= ))


119898 isin N) and (119906119898

119898 isin N)


119898isin 119879(119906

119898) Then forall

119898ge2 119909119898

isin [1 2] 2 isin





following


0isin 119883




0isin 119883





119909119910isin1198838119863(119879(119909)


119901(119906119882) sup



119909119910isin1198838119863(119879(119909) 119879(119910)) le 1205820119901(119909 119910) Letting 1199090 = 2


[1 2] = 119880 119879(1199100) = (4 6) = 119882 sup119906isin[12]119901(119906 (4 6)) =

sup119906isin[12](4 minus 119906)

4= 34 = 81 and sup

119908isin(46)119901([1 2] 119908) =




119862A)


119862A whereJ



119862A)whenJ119862A = P


119862A





119860 = 1198601 cup1198602 1198601 = (0 2] 1198602 = [4 6) (77)

Let 119901 1198832


119901 (119906 119908) =

0 if 119860 cap 119906 119908 = 119906 119908

120574 if 119860 cap 119906 119908 = 119906 119908

(78)

and letJ11 = P


119879 (119906) =

1198602 for 119906 isin (0 3)

119860 for 119906 = 3

1198601 for 119906 isin (3 6)

(79)


Section 11(2) (119883 119879) is a (D



119909119910isin119883119863

119871minusP1112119883 (119879(119909) 119879(119910)) le 120582119901(119909


and maxsup119906isin119879(119909)

119901(119906 119879(119910)) sup119908isin119879(119910)



exist119910isin119879(119909)



119901 (1199090 1199100) = 119901 (1199090 119879 (1199090))

=

0 if 1199090 isin 1198601

120574 for 1199090 isin (0 3) 1198601

(80)

Therefore 119901(1199090 1199100) lt 119901(1199090 119879(1199090)) + 120573



119901 (1199090 1199100) = 119901 (1199090 119879 (1199090))

=

0 if 1199090 isin 1198602

120574 for 1199090 isin (3 6) 1198602

(81)







sup119899gt119898

119901(119908119898

119908119899


lim119898rarrinfin

119901(119908119908119898


(79)forall119898⩾1 119908

119898

isin 119879(119908119898minus1



(119908119898119898isin0cupN)




119898 isin N) sub 119879[2]

(119883) = 119860



119898 isin N) and (119906119898


119898isin 119879(119906


exist1198980isinN

forall119898⩾1198980

119909119898



and Fix(119879[2]) = 119860

(6) For (119883P11)P11 = 119901 J

11 = P11 and



isin

119883 1199081isin 119879(119908

0) and 119908

2= 119906 isin 119879(119908


and forall119898⩾3 119908

119898

= 119906 then sequence (119908119898




1198832


119901 (119906 119908) =

0 if 119906 ge 119908

(119906 minus 119908)2 if 119906 lt 119908

(82)


119879 (119906) = ([0 3) cup (3 6]) 119906 for 119906 isin [0 6] (83)

Let

119864 = [0 3) cup (3 6] (84)


2rarr [0infin) is

of the form

119869 (119906 119908) =

119901 (119906 119908) if 119864 cap 119906 119908 = 119906 119908

120583 if 119864 cap 119906 119908 = 119906 119908

(85)


119869(0 6) = 36 and 119869(6 0) = 0(2)J

21 = 119869 isin J119871(119883P21)

capJ119877(119883P21)

SeeTheorem 14




[0 1) and

119863(119880119882) = maxsup119906isin119880

119869 (119906119882) sup119908isin119882

119869 (119880 119908)

119880119882 isin 2119883

(86)




119906

cup 119882119906and

inf119908isin119882

119869 (119906 119908)

le

inf119908isin119882119906119902 (119906 119908) = 0 if 119882

119906

= 119908 isin 119882 119906 ge 119908 =

inf119908isin119882119906

(119906 minus 119908)2= 0 if 119882

119906= 119908 isin 119882 119906 lt 119908 =

(87)


cup 119880119908and

inf119906isin119880

119869 (119906 119908)

le

inf119906isin119880119908119902 (119906 119908) = 0 if 119880

119906

= 119906 isin 119880 119906 ge 119908 =

inf119906isin119880119908

(119906 minus 119908)2= 0 if 119880

119906= 119906 isin 119880 119906 lt 119908 =

(88)

By (86) 2119863(119879(119909) 119879(119910)) = 0 le 120582119869(119909 119910)

Case 2 If 119909 = 119910 = 3 then 119869(119909 119910) = 120583 and 119879(119909) =


2119863(119880119880) = 0 le 120582119869(119909 119910)



inf119908isin119882

119869(119906 119908) =


119869(119906 119908) =


inf119906isin119880


119908

cup 119880119908and

inf119906isin119880

119869 (119906 119908)

le

inf119906isin119880119908119902 (119906 119908) = 0 if 119880

119906

= 119906 isin 119880 119906 ge 119908 =

inf119906isin119880119908

(119906 minus 119908)2= 0 if 119880

119906= 119906 isin 119880 119906 lt 119908 =

(89)

Thus 2119863(119879(119909) 119879(119910)) = 0 ⩽ 120582119869(119909 119910)


(3 6]) 119910 = 119882 sub 119864 and sup119906isin119880

inf119908isin119882

119869(119906 119908) = 0 sincefor 119906 isin 119880

inf119908isin119882

119869 (119906 119908)

le

inf119908isin119882119906119902 (119906 119908) = 0 if 119882

119906

= 119908 isin 119882 119906 ge 119908 =

inf119908isin119882119906

(119906 minus 119908)2= 0 if 119882

119906= 119908 isin 119882 119906 lt 119908 =

(90)


inf119906isin119880

119869(119906 119908) = 0 since inf119906isin119880

119869(119906 119908) = 119869(119908119908) = 0 for 119908 isin 119882 Thus 2119863(119879(119909) 119879(119910)) = 0 le 120582119869(119909 119910)



exist119910isin119879(119909)

119869(119909




inf119908isin119882

119869(1199090 119908) = 0 since

inf119908isin119882

119869 (1199090 119908)

le

inf119908isin1198821199090119902 (1199090 119908) = 0 if 119882

1199090 = 119908 isin 119882 1199090 ge 119908 =

inf119908isin1198821199090

(1199090 minus 119908)2= 0 if 119882

1199090= 119908 isin 119882 1199090 lt 119908 =

(91)



12)

then 119869(1199090 1199100) lt 119869(1199090 119879(1199090)) + 120574



119869(1199090 1199100) lt 119869(1199090 119879(1199090)) + 120574




119869(1199090 119879(1199090)) + 120574


1199080


119898



sup119899gt119898

119869(119908119898

119908119899


lim119898rarrinfin

119869(119908 119908119898



1isin 119879(119908

0) = ([0 3) cup

(3 6]) 1199080 and forall

119898ge2 119908119898

isin 119879(119908119898minus1


(119908119898119898isin0cupN)

Case 2 If 1199080= 3 then 119908

1isin 119879(119908

0) = [0 3) cup (3 6] 1199082

isin

119879(1199081) = ([0 3) cup (3 6]) 119908

1 and forall

119898⩾3 119908119898

isin 119879(119908119898minus1

) sub




0isin 119883 and



119898rarrinfinsup

119899gt119898119869(119908

119898

119908119899

) =

0 of (119908119898


[2]) is a left P


119898 119898 isin N) sub 119879

[2](119883) =



119898 isin N) and (119906119898

119898 isin N)


119898isin 119879(119906


exist1198980isinN

forall119898ge1198980

119909119898


(119909119898 119898isinN)and 6 isin 119879

[2](6)

(7) ForP21 = 119901J



that 6 isin 119879[2]


0isin 119883






11 = 119901


119879 (119906) =

4 for 119906 isin (0 3)

2 for 119906 isin [3 6) (92)



119909119910isin119883119863

119871minusP111119883 (119879(119909) 119879(119910)) =

max119901(119879(119909) 119879(119910)) 119901(119879(119910) 119879(119909)) le 120582119901(119909 119910) and Fix(119879) =






119898+1= 119879(119908

119898


sup119899gt119898

119901(119908119898

119908119899

) =


119898



(119908119898119898isin0cupN)




119898 isin N) sub

119879[2]



119898 isin N) and (119906119898

119898 isin N)


119898= 119879

[2](119906

119898) then by (77) (78) and (92)



(2) isin

LIM119871minusP11(119909119898 119898isinN)

and 4 = 119879[2]





119883119860 then 119901(119906 119908) = 119901(119908 119906) = 120574 gt 0(6) For P

11 = 119901 (119883 119879) and J11 = P





119889


119889



(119883 119889) rarr (CB(119883)119867119889











References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










le (1205821205720

1198621205720

)1198691205720

(119879[2119896minus1]

(V) 119879[2119896minus1](119879 (V)))

le (1205821205720

1198621205720

) sdot119863119871minusJ119862A1205781198831205720

(119879[2119896minus1]

(V) 119879[2119896minus1](119879 (V)))

le (1205821205720

1198621205720

)

2

1198691205720

(119879[2119896minus2]


le (1205821205720

1198621205720

)

2119896

1198691205720

(V 119879 (V)) lt 1198691205720

(V 119879 (V))

(59)



120572(V 119879 (V)) = 0 (60)




119879[119896]

(V) = 119879[2119896]


0 lt 1198691205720

(119879 (V) V) = 1198691205720

(119879[119896+1]

(V) 119879[2119896](V))

le

119896minus2sum

119898=1119862119898

12057201198691205720

(119879[119896+119898]

(V) 119879[119896+119898+1](V))

+119862119896minus21205720

1198691205720

(119879[2119896minus1]

(V) 119879[2119896](V))

le

119896minus2sum

119898=1119862119898

1205720sdot 119863

119871minusJ119862A1205781198831205720

(119879[119896+119898]

(V) 119879[119896+119898+1](V))

+119862119896minus21205720

sdot 119863119871minusJ119862A1205781198831205720

(119879[2119896minus1]

(V) 119879[2119896](V))

le

119896minus2sum

119898=1119862119898

1205720(

1205821205720

1198621205720

)

119896+119898

1198691205720

(V 119879 (V))

+119862119896minus21205720

(1205821205720

1198621205720

)

2119896minus1

1198691205720

(V 119879 (V)) = 0

(61)



120572(119879 (V) V) = 0 (62)






)


120572 120572 isin A is


120572(119879(V) V) gt



120572(119879(V) V) gt 0 or 119869

120572(V 119879(V)) gt 0 However by


120572(V V) = 0 In fact


120572(V V) le 119862

120572[119869

120572(V 119879(V))+119869

120572(119879(V) V)] = 0




1199011205720(119906 119908) gt 0 or 119901

1205720(119908 119906) gt 0 By applying


1198691205720(119906 119908) gt 0 or 119869

1205720(119908 119906) gt


exist1205720isinA

[1198691205720

(119906 119908) gt 0 1198691205720

(119906 119908)

= 1198691205720

(119879 (119906) 119879 (119908)) le119863119871minusJ119862A1205781198831205720

(119879 (119906) 119879 (119908))

le(1205821205720

1198621205720

)1198691205720

(119906 119908) lt 1198691205720

(119906 119908)] or [1198691205720

(119908 119906)

gt 0 1198691205720

(119908 119906) = 1198691205720

(119879 (119908) 119879 (119906))

le119863119871minusJ119862A1205781198831205720

(119879 (119908) 119879 (119906)) le(1205821205720

1198621205720

)1198691205720

(119908 119906)

lt 1198691205720

(119908 119906)]

(64)




rarr [0infin)

be of the form

119901 (119906 119908)

=

0 if 119906 ge 119908 119906 119908 cap (0 6) = 119906 119908

(119908 minus 119906)4 if 119906 lt 119908 119906 119908 cap (0 6) = 119906 119908

120574 if 119906 119908 cap (0 6) = 119906 119908

(65)


space In fact





8(119908 minus V)4 = 8[119901(119906 V) + 119901(V 119908)]


(119908 minus 119906)4 and 119891(V0) = min



and V0 = (119906 + 119908)2


119906119908isin(06)119906lt119908(119908 minus 119906)4

=


119906leVle1199088[119901(119906 V) + 119901(V 119908)] =

sup119906119908isin(06)119906lt119908min


(6 minus 3)4] = 8[81 + 81] = 1296




4lt

8(V minus 119906)4= 8[119901(119906 V) + 119901(V 119908)]


1296 le 8[119901(119906 V) + 119901(V 119908)] = 8[120574 + 120574]


8120574 le 8[119901(119906 V) + 119901(V 119908)]



119901(119906119908) = 119901(119908 119906) does not hold




119898= 3





(119906119898 119898isinN)= [0 3]



119901 (119906 119908) =

0 if 119860 cap 119906 119908 = 119906 119908

120574 if 119860 cap 119906 119908 = 119906 119908

(67)









119901(119906 119906) = 0 does not hold(3)P


(119906119898 119898isinN)= LIM119877minusP11

(119906119898 119898isinN)= 119860 for





119901 (119906 119908) =

0 if 119906 ge 119908

(119908 minus 119906)3 if 119906 lt 119908

(68)






4119901(V 119908) = 4[119901(119906 V) + 119901(V 119908)]


119891(V0) = min119906⩽V⩽119908119891(V) = (119908 minus 119906)

3 where V0 = (119906 + 119908)2 is


4(119908 minus 119906)[1199082+ 119908119906 + 119906

2+ 3V2 minus 3V(119908 + 119906)]



3lt 4(V minus 119906)

3= 4119901(119906 V) = 4[119901(119906

V) + 119901(V 119908)]



119901(119906119908) = 119901(119908 119906) does not hold




(119906119898 119898isinN)= [2 6] and

LIM119877minusP41(119906119898 119898isinN)

= [1 2] for sequence (119906119898



1198832


119901 (119906 119908) =

0 if 119906 ge 119908

(119906 minus 119908)2 if 119906 lt 119908

(69)

Let

119864 = [0 3) cup (3 6] (70)


2rarr [0infin) is

of the form

119869 (119906 119908) =

119901 (119906 119908) if 119864 cap 119906 119908 = 119906 119908

120583 if 119864 cap 119906 119908 = 119906 119908

(71)


36 and 119869(6 0) = 0(2)J

21 = 119869 isin J119871(119883P21)

capJ119877(119883P21)

SeeTheorem 14




119901 (119906 119908)

=

0 if 119906 ge 119908 119906 119908 cap (0 6) = 119906 119908

(119908 minus 119906)4 if 119906 lt 119908 119906 119908 cap (0 6) = 119906 119908

120574 if 119906 119908 cap (0 6) = 119906 119908

(72)


119879 (119906) =

[1 2] if 119906 isin [0 3) cup (4 6]

(4 6) if 119906 isin [3 4] (73)


Let119864 = [0 3) cup (4 6] (74)


119869 (119906 119908) =

119901 (119906 119908) if 119864 cap 119906 119908 = 119906 119908

120574 if 119864 cap 119906 119908 = 119906 119908

(75)

(1) (119883P81) where P




(2) (119883 119879) is a (119863 = D119871minusJ8112119883 = D

119877minusJ8112119883 120582 isin [2048


119909119910isin1198838 sdot

119863(119879(119909) 119879(119910)) le 120582119869(119909 119910) where

119863 (119880119882) = maxsup119906isin119880

119869 (119906119882) sup119908isin119882

119869 (119880 119908)

119880119882 isin 2119883

(76)



119864 and sup119906isin119880

inf119908isin119880

119869(119906 119908) = sup119906isin119880

119869(119906 119906) = 119901(119906 119906) =

0 = 0 Thus 4119863(119879(119909) 119879(119910)) = 0 le 120582119869(119909 119910)



inf119908isin119882

119869(119906 119908) =

sup119906isin119880

inf119908isin119882

(119908 minus 119906)4 = sup

119906isin119880(4 minus 119906)

4= 81 and

sup119908isin119882

inf119906isin119880

119869(119906 119908) = sup119908isin119882

inf119906isin119880

(119908 minus 119906)4 =

sup119908isin119882





119869(119906 119908) = sup119906isin119880

inf119908isin119882

119901(119906 119908) =

sup119908isin119882

inf119906isin119880

119869(119906 119908) = sup119908isin119882

inf119906isin119880

119901(119906 119908) = 0Therefore 8119863(119879(119909) 119879(119910)) = 0 le 120582119869(119909 119910)




exist119910isin119879(119909)

119869(119909




120573isin(0infin)119869(1199090

1199100) lt 119869(1199090 119879(1199090)) + 120573



120573isin(0infin)119869(1199090 1199100) lt 119869(1199090 119879(1199090)) + 120573


120573isin(0infin)119869(1199090 1199100) lt

119869(1199090 119879(1199090)) + 120573



119869(1199090 1199100) lt 119869(1199090 119879(1199090)) +

120573


120573isin(0infin)119869(1199090 1199100) lt 119869(1199090 119879(1199090)) + 120573


1199080isin 119883 In fact if1199080

isin 119883 and (119908119898


119898+1isin 119879(119908

119898


sup119899gt119898

119869(119908119898

119908119899

) =


sup119899gt119898

119869(119908119899

119908119898

) = 0) then forall119898ge2 119908

119898



119901(119908119908119898

) =


119898rarrinfin119901(119908

119898



119869(119908 119908119898

) =


119898rarrinfin119869(119908

119898





(119909119898 119898isinN)= (LIM119877minusP81

(119909119898 119898isinN)= ))


119898 isin N) and (119906119898

119898 isin N)


119898isin 119879(119906

119898) Then forall

119898ge2 119909119898

isin [1 2] 2 isin





following


0isin 119883




0isin 119883





119909119910isin1198838119863(119879(119909)


119901(119906119882) sup



119909119910isin1198838119863(119879(119909) 119879(119910)) le 1205820119901(119909 119910) Letting 1199090 = 2


[1 2] = 119880 119879(1199100) = (4 6) = 119882 sup119906isin[12]119901(119906 (4 6)) =

sup119906isin[12](4 minus 119906)

4= 34 = 81 and sup

119908isin(46)119901([1 2] 119908) =




119862A)


119862A whereJ



119862A)whenJ119862A = P


119862A





119860 = 1198601 cup1198602 1198601 = (0 2] 1198602 = [4 6) (77)

Let 119901 1198832


119901 (119906 119908) =

0 if 119860 cap 119906 119908 = 119906 119908

120574 if 119860 cap 119906 119908 = 119906 119908

(78)

and letJ11 = P


119879 (119906) =

1198602 for 119906 isin (0 3)

119860 for 119906 = 3

1198601 for 119906 isin (3 6)

(79)


Section 11(2) (119883 119879) is a (D



119909119910isin119883119863

119871minusP1112119883 (119879(119909) 119879(119910)) le 120582119901(119909


and maxsup119906isin119879(119909)

119901(119906 119879(119910)) sup119908isin119879(119910)



exist119910isin119879(119909)



119901 (1199090 1199100) = 119901 (1199090 119879 (1199090))

=

0 if 1199090 isin 1198601

120574 for 1199090 isin (0 3) 1198601

(80)

Therefore 119901(1199090 1199100) lt 119901(1199090 119879(1199090)) + 120573



119901 (1199090 1199100) = 119901 (1199090 119879 (1199090))

=

0 if 1199090 isin 1198602

120574 for 1199090 isin (3 6) 1198602

(81)







sup119899gt119898

119901(119908119898

119908119899


lim119898rarrinfin

119901(119908119908119898


(79)forall119898⩾1 119908

119898

isin 119879(119908119898minus1



(119908119898119898isin0cupN)




119898 isin N) sub 119879[2]

(119883) = 119860



119898 isin N) and (119906119898


119898isin 119879(119906


exist1198980isinN

forall119898⩾1198980

119909119898



and Fix(119879[2]) = 119860

(6) For (119883P11)P11 = 119901 J

11 = P11 and



isin

119883 1199081isin 119879(119908

0) and 119908

2= 119906 isin 119879(119908


and forall119898⩾3 119908

119898

= 119906 then sequence (119908119898




1198832


119901 (119906 119908) =

0 if 119906 ge 119908

(119906 minus 119908)2 if 119906 lt 119908

(82)


119879 (119906) = ([0 3) cup (3 6]) 119906 for 119906 isin [0 6] (83)

Let

119864 = [0 3) cup (3 6] (84)


2rarr [0infin) is

of the form

119869 (119906 119908) =

119901 (119906 119908) if 119864 cap 119906 119908 = 119906 119908

120583 if 119864 cap 119906 119908 = 119906 119908

(85)


119869(0 6) = 36 and 119869(6 0) = 0(2)J

21 = 119869 isin J119871(119883P21)

capJ119877(119883P21)

SeeTheorem 14




[0 1) and

119863(119880119882) = maxsup119906isin119880

119869 (119906119882) sup119908isin119882

119869 (119880 119908)

119880119882 isin 2119883

(86)




119906

cup 119882119906and

inf119908isin119882

119869 (119906 119908)

le

inf119908isin119882119906119902 (119906 119908) = 0 if 119882

119906

= 119908 isin 119882 119906 ge 119908 =

inf119908isin119882119906

(119906 minus 119908)2= 0 if 119882

119906= 119908 isin 119882 119906 lt 119908 =

(87)


cup 119880119908and

inf119906isin119880

119869 (119906 119908)

le

inf119906isin119880119908119902 (119906 119908) = 0 if 119880

119906

= 119906 isin 119880 119906 ge 119908 =

inf119906isin119880119908

(119906 minus 119908)2= 0 if 119880

119906= 119906 isin 119880 119906 lt 119908 =

(88)

By (86) 2119863(119879(119909) 119879(119910)) = 0 le 120582119869(119909 119910)

Case 2 If 119909 = 119910 = 3 then 119869(119909 119910) = 120583 and 119879(119909) =


2119863(119880119880) = 0 le 120582119869(119909 119910)



inf119908isin119882

119869(119906 119908) =


119869(119906 119908) =


inf119906isin119880


119908

cup 119880119908and

inf119906isin119880

119869 (119906 119908)

le

inf119906isin119880119908119902 (119906 119908) = 0 if 119880

119906

= 119906 isin 119880 119906 ge 119908 =

inf119906isin119880119908

(119906 minus 119908)2= 0 if 119880

119906= 119906 isin 119880 119906 lt 119908 =

(89)

Thus 2119863(119879(119909) 119879(119910)) = 0 ⩽ 120582119869(119909 119910)


(3 6]) 119910 = 119882 sub 119864 and sup119906isin119880

inf119908isin119882

119869(119906 119908) = 0 sincefor 119906 isin 119880

inf119908isin119882

119869 (119906 119908)

le

inf119908isin119882119906119902 (119906 119908) = 0 if 119882

119906

= 119908 isin 119882 119906 ge 119908 =

inf119908isin119882119906

(119906 minus 119908)2= 0 if 119882

119906= 119908 isin 119882 119906 lt 119908 =

(90)


inf119906isin119880

119869(119906 119908) = 0 since inf119906isin119880

119869(119906 119908) = 119869(119908119908) = 0 for 119908 isin 119882 Thus 2119863(119879(119909) 119879(119910)) = 0 le 120582119869(119909 119910)



exist119910isin119879(119909)

119869(119909




inf119908isin119882

119869(1199090 119908) = 0 since

inf119908isin119882

119869 (1199090 119908)

le

inf119908isin1198821199090119902 (1199090 119908) = 0 if 119882

1199090 = 119908 isin 119882 1199090 ge 119908 =

inf119908isin1198821199090

(1199090 minus 119908)2= 0 if 119882

1199090= 119908 isin 119882 1199090 lt 119908 =

(91)



12)

then 119869(1199090 1199100) lt 119869(1199090 119879(1199090)) + 120574



119869(1199090 1199100) lt 119869(1199090 119879(1199090)) + 120574




119869(1199090 119879(1199090)) + 120574


1199080


119898



sup119899gt119898

119869(119908119898

119908119899


lim119898rarrinfin

119869(119908 119908119898



1isin 119879(119908

0) = ([0 3) cup

(3 6]) 1199080 and forall

119898ge2 119908119898

isin 119879(119908119898minus1


(119908119898119898isin0cupN)

Case 2 If 1199080= 3 then 119908

1isin 119879(119908

0) = [0 3) cup (3 6] 1199082

isin

119879(1199081) = ([0 3) cup (3 6]) 119908

1 and forall

119898⩾3 119908119898

isin 119879(119908119898minus1

) sub




0isin 119883 and



119898rarrinfinsup

119899gt119898119869(119908

119898

119908119899

) =

0 of (119908119898


[2]) is a left P


119898 119898 isin N) sub 119879

[2](119883) =



119898 isin N) and (119906119898

119898 isin N)


119898isin 119879(119906


exist1198980isinN

forall119898ge1198980

119909119898


(119909119898 119898isinN)and 6 isin 119879

[2](6)

(7) ForP21 = 119901J



that 6 isin 119879[2]


0isin 119883






11 = 119901


119879 (119906) =

4 for 119906 isin (0 3)

2 for 119906 isin [3 6) (92)



119909119910isin119883119863

119871minusP111119883 (119879(119909) 119879(119910)) =

max119901(119879(119909) 119879(119910)) 119901(119879(119910) 119879(119909)) le 120582119901(119909 119910) and Fix(119879) =






119898+1= 119879(119908

119898


sup119899gt119898

119901(119908119898

119908119899

) =


119898



(119908119898119898isin0cupN)




119898 isin N) sub

119879[2]



119898 isin N) and (119906119898

119898 isin N)


119898= 119879

[2](119906

119898) then by (77) (78) and (92)



(2) isin

LIM119871minusP11(119909119898 119898isinN)

and 4 = 119879[2]





119883119860 then 119901(119906 119908) = 119901(119908 119906) = 120574 gt 0(6) For P

11 = 119901 (119883 119879) and J11 = P





119889


119889



(119883 119889) rarr (CB(119883)119867119889











References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












4lt

8(V minus 119906)4= 8[119901(119906 V) + 119901(V 119908)]


1296 le 8[119901(119906 V) + 119901(V 119908)] = 8[120574 + 120574]


8120574 le 8[119901(119906 V) + 119901(V 119908)]



119901(119906119908) = 119901(119908 119906) does not hold




119898= 3





(119906119898 119898isinN)= [0 3]



119901 (119906 119908) =

0 if 119860 cap 119906 119908 = 119906 119908

120574 if 119860 cap 119906 119908 = 119906 119908

(67)









119901(119906 119906) = 0 does not hold(3)P


(119906119898 119898isinN)= LIM119877minusP11

(119906119898 119898isinN)= 119860 for





119901 (119906 119908) =

0 if 119906 ge 119908

(119908 minus 119906)3 if 119906 lt 119908

(68)






4119901(V 119908) = 4[119901(119906 V) + 119901(V 119908)]


119891(V0) = min119906⩽V⩽119908119891(V) = (119908 minus 119906)

3 where V0 = (119906 + 119908)2 is


4(119908 minus 119906)[1199082+ 119908119906 + 119906

2+ 3V2 minus 3V(119908 + 119906)]



3lt 4(V minus 119906)

3= 4119901(119906 V) = 4[119901(119906

V) + 119901(V 119908)]



119901(119906119908) = 119901(119908 119906) does not hold




(119906119898 119898isinN)= [2 6] and

LIM119877minusP41(119906119898 119898isinN)

= [1 2] for sequence (119906119898



1198832


119901 (119906 119908) =

0 if 119906 ge 119908

(119906 minus 119908)2 if 119906 lt 119908

(69)

Let

119864 = [0 3) cup (3 6] (70)


2rarr [0infin) is

of the form

119869 (119906 119908) =

119901 (119906 119908) if 119864 cap 119906 119908 = 119906 119908

120583 if 119864 cap 119906 119908 = 119906 119908

(71)


36 and 119869(6 0) = 0(2)J

21 = 119869 isin J119871(119883P21)

capJ119877(119883P21)

SeeTheorem 14




119901 (119906 119908)

=

0 if 119906 ge 119908 119906 119908 cap (0 6) = 119906 119908

(119908 minus 119906)4 if 119906 lt 119908 119906 119908 cap (0 6) = 119906 119908

120574 if 119906 119908 cap (0 6) = 119906 119908

(72)


119879 (119906) =

[1 2] if 119906 isin [0 3) cup (4 6]

(4 6) if 119906 isin [3 4] (73)


Let119864 = [0 3) cup (4 6] (74)


119869 (119906 119908) =

119901 (119906 119908) if 119864 cap 119906 119908 = 119906 119908

120574 if 119864 cap 119906 119908 = 119906 119908

(75)

(1) (119883P81) where P




(2) (119883 119879) is a (119863 = D119871minusJ8112119883 = D

119877minusJ8112119883 120582 isin [2048


119909119910isin1198838 sdot

119863(119879(119909) 119879(119910)) le 120582119869(119909 119910) where

119863 (119880119882) = maxsup119906isin119880

119869 (119906119882) sup119908isin119882

119869 (119880 119908)

119880119882 isin 2119883

(76)



119864 and sup119906isin119880

inf119908isin119880

119869(119906 119908) = sup119906isin119880

119869(119906 119906) = 119901(119906 119906) =

0 = 0 Thus 4119863(119879(119909) 119879(119910)) = 0 le 120582119869(119909 119910)



inf119908isin119882

119869(119906 119908) =

sup119906isin119880

inf119908isin119882

(119908 minus 119906)4 = sup

119906isin119880(4 minus 119906)

4= 81 and

sup119908isin119882

inf119906isin119880

119869(119906 119908) = sup119908isin119882

inf119906isin119880

(119908 minus 119906)4 =

sup119908isin119882





119869(119906 119908) = sup119906isin119880

inf119908isin119882

119901(119906 119908) =

sup119908isin119882

inf119906isin119880

119869(119906 119908) = sup119908isin119882

inf119906isin119880

119901(119906 119908) = 0Therefore 8119863(119879(119909) 119879(119910)) = 0 le 120582119869(119909 119910)




exist119910isin119879(119909)

119869(119909




120573isin(0infin)119869(1199090

1199100) lt 119869(1199090 119879(1199090)) + 120573



120573isin(0infin)119869(1199090 1199100) lt 119869(1199090 119879(1199090)) + 120573


120573isin(0infin)119869(1199090 1199100) lt

119869(1199090 119879(1199090)) + 120573



119869(1199090 1199100) lt 119869(1199090 119879(1199090)) +

120573


120573isin(0infin)119869(1199090 1199100) lt 119869(1199090 119879(1199090)) + 120573


1199080isin 119883 In fact if1199080

isin 119883 and (119908119898


119898+1isin 119879(119908

119898


sup119899gt119898

119869(119908119898

119908119899

) =


sup119899gt119898

119869(119908119899

119908119898

) = 0) then forall119898ge2 119908

119898



119901(119908119908119898

) =


119898rarrinfin119901(119908

119898



119869(119908 119908119898

) =


119898rarrinfin119869(119908

119898





(119909119898 119898isinN)= (LIM119877minusP81

(119909119898 119898isinN)= ))


119898 isin N) and (119906119898

119898 isin N)


119898isin 119879(119906

119898) Then forall

119898ge2 119909119898

isin [1 2] 2 isin





following


0isin 119883




0isin 119883





119909119910isin1198838119863(119879(119909)


119901(119906119882) sup



119909119910isin1198838119863(119879(119909) 119879(119910)) le 1205820119901(119909 119910) Letting 1199090 = 2


[1 2] = 119880 119879(1199100) = (4 6) = 119882 sup119906isin[12]119901(119906 (4 6)) =

sup119906isin[12](4 minus 119906)

4= 34 = 81 and sup

119908isin(46)119901([1 2] 119908) =




119862A)


119862A whereJ



119862A)whenJ119862A = P


119862A





119860 = 1198601 cup1198602 1198601 = (0 2] 1198602 = [4 6) (77)

Let 119901 1198832


119901 (119906 119908) =

0 if 119860 cap 119906 119908 = 119906 119908

120574 if 119860 cap 119906 119908 = 119906 119908

(78)

and letJ11 = P


119879 (119906) =

1198602 for 119906 isin (0 3)

119860 for 119906 = 3

1198601 for 119906 isin (3 6)

(79)


Section 11(2) (119883 119879) is a (D



119909119910isin119883119863

119871minusP1112119883 (119879(119909) 119879(119910)) le 120582119901(119909


and maxsup119906isin119879(119909)

119901(119906 119879(119910)) sup119908isin119879(119910)



exist119910isin119879(119909)



119901 (1199090 1199100) = 119901 (1199090 119879 (1199090))

=

0 if 1199090 isin 1198601

120574 for 1199090 isin (0 3) 1198601

(80)

Therefore 119901(1199090 1199100) lt 119901(1199090 119879(1199090)) + 120573



119901 (1199090 1199100) = 119901 (1199090 119879 (1199090))

=

0 if 1199090 isin 1198602

120574 for 1199090 isin (3 6) 1198602

(81)







sup119899gt119898

119901(119908119898

119908119899


lim119898rarrinfin

119901(119908119908119898


(79)forall119898⩾1 119908

119898

isin 119879(119908119898minus1



(119908119898119898isin0cupN)




119898 isin N) sub 119879[2]

(119883) = 119860



119898 isin N) and (119906119898


119898isin 119879(119906


exist1198980isinN

forall119898⩾1198980

119909119898



and Fix(119879[2]) = 119860

(6) For (119883P11)P11 = 119901 J

11 = P11 and



isin

119883 1199081isin 119879(119908

0) and 119908

2= 119906 isin 119879(119908


and forall119898⩾3 119908

119898

= 119906 then sequence (119908119898




1198832


119901 (119906 119908) =

0 if 119906 ge 119908

(119906 minus 119908)2 if 119906 lt 119908

(82)


119879 (119906) = ([0 3) cup (3 6]) 119906 for 119906 isin [0 6] (83)

Let

119864 = [0 3) cup (3 6] (84)


2rarr [0infin) is

of the form

119869 (119906 119908) =

119901 (119906 119908) if 119864 cap 119906 119908 = 119906 119908

120583 if 119864 cap 119906 119908 = 119906 119908

(85)


119869(0 6) = 36 and 119869(6 0) = 0(2)J

21 = 119869 isin J119871(119883P21)

capJ119877(119883P21)

SeeTheorem 14




[0 1) and

119863(119880119882) = maxsup119906isin119880

119869 (119906119882) sup119908isin119882

119869 (119880 119908)

119880119882 isin 2119883

(86)




119906

cup 119882119906and

inf119908isin119882

119869 (119906 119908)

le

inf119908isin119882119906119902 (119906 119908) = 0 if 119882

119906

= 119908 isin 119882 119906 ge 119908 =

inf119908isin119882119906

(119906 minus 119908)2= 0 if 119882

119906= 119908 isin 119882 119906 lt 119908 =

(87)


cup 119880119908and

inf119906isin119880

119869 (119906 119908)

le

inf119906isin119880119908119902 (119906 119908) = 0 if 119880

119906

= 119906 isin 119880 119906 ge 119908 =

inf119906isin119880119908

(119906 minus 119908)2= 0 if 119880

119906= 119906 isin 119880 119906 lt 119908 =

(88)

By (86) 2119863(119879(119909) 119879(119910)) = 0 le 120582119869(119909 119910)

Case 2 If 119909 = 119910 = 3 then 119869(119909 119910) = 120583 and 119879(119909) =


2119863(119880119880) = 0 le 120582119869(119909 119910)



inf119908isin119882

119869(119906 119908) =


119869(119906 119908) =


inf119906isin119880


119908

cup 119880119908and

inf119906isin119880

119869 (119906 119908)

le

inf119906isin119880119908119902 (119906 119908) = 0 if 119880

119906

= 119906 isin 119880 119906 ge 119908 =

inf119906isin119880119908

(119906 minus 119908)2= 0 if 119880

119906= 119906 isin 119880 119906 lt 119908 =

(89)

Thus 2119863(119879(119909) 119879(119910)) = 0 ⩽ 120582119869(119909 119910)


(3 6]) 119910 = 119882 sub 119864 and sup119906isin119880

inf119908isin119882

119869(119906 119908) = 0 sincefor 119906 isin 119880

inf119908isin119882

119869 (119906 119908)

le

inf119908isin119882119906119902 (119906 119908) = 0 if 119882

119906

= 119908 isin 119882 119906 ge 119908 =

inf119908isin119882119906

(119906 minus 119908)2= 0 if 119882

119906= 119908 isin 119882 119906 lt 119908 =

(90)


inf119906isin119880

119869(119906 119908) = 0 since inf119906isin119880

119869(119906 119908) = 119869(119908119908) = 0 for 119908 isin 119882 Thus 2119863(119879(119909) 119879(119910)) = 0 le 120582119869(119909 119910)



exist119910isin119879(119909)

119869(119909




inf119908isin119882

119869(1199090 119908) = 0 since

inf119908isin119882

119869 (1199090 119908)

le

inf119908isin1198821199090119902 (1199090 119908) = 0 if 119882

1199090 = 119908 isin 119882 1199090 ge 119908 =

inf119908isin1198821199090

(1199090 minus 119908)2= 0 if 119882

1199090= 119908 isin 119882 1199090 lt 119908 =

(91)



12)

then 119869(1199090 1199100) lt 119869(1199090 119879(1199090)) + 120574



119869(1199090 1199100) lt 119869(1199090 119879(1199090)) + 120574




119869(1199090 119879(1199090)) + 120574


1199080


119898



sup119899gt119898

119869(119908119898

119908119899


lim119898rarrinfin

119869(119908 119908119898



1isin 119879(119908

0) = ([0 3) cup

(3 6]) 1199080 and forall

119898ge2 119908119898

isin 119879(119908119898minus1


(119908119898119898isin0cupN)

Case 2 If 1199080= 3 then 119908

1isin 119879(119908

0) = [0 3) cup (3 6] 1199082

isin

119879(1199081) = ([0 3) cup (3 6]) 119908

1 and forall

119898⩾3 119908119898

isin 119879(119908119898minus1

) sub




0isin 119883 and



119898rarrinfinsup

119899gt119898119869(119908

119898

119908119899

) =

0 of (119908119898


[2]) is a left P


119898 119898 isin N) sub 119879

[2](119883) =



119898 isin N) and (119906119898

119898 isin N)


119898isin 119879(119906


exist1198980isinN

forall119898ge1198980

119909119898


(119909119898 119898isinN)and 6 isin 119879

[2](6)

(7) ForP21 = 119901J



that 6 isin 119879[2]


0isin 119883






11 = 119901


119879 (119906) =

4 for 119906 isin (0 3)

2 for 119906 isin [3 6) (92)



119909119910isin119883119863

119871minusP111119883 (119879(119909) 119879(119910)) =

max119901(119879(119909) 119879(119910)) 119901(119879(119910) 119879(119909)) le 120582119901(119909 119910) and Fix(119879) =






119898+1= 119879(119908

119898


sup119899gt119898

119901(119908119898

119908119899

) =


119898



(119908119898119898isin0cupN)




119898 isin N) sub

119879[2]



119898 isin N) and (119906119898

119898 isin N)


119898= 119879

[2](119906

119898) then by (77) (78) and (92)



(2) isin

LIM119871minusP11(119909119898 119898isinN)

and 4 = 119879[2]





119883119860 then 119901(119906 119908) = 119901(119908 119906) = 120574 gt 0(6) For P

11 = 119901 (119883 119879) and J11 = P





119889


119889



(119883 119889) rarr (CB(119883)119867119889











References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Let119864 = [0 3) cup (4 6] (74)


119869 (119906 119908) =

119901 (119906 119908) if 119864 cap 119906 119908 = 119906 119908

120574 if 119864 cap 119906 119908 = 119906 119908

(75)

(1) (119883P81) where P




(2) (119883 119879) is a (119863 = D119871minusJ8112119883 = D

119877minusJ8112119883 120582 isin [2048


119909119910isin1198838 sdot

119863(119879(119909) 119879(119910)) le 120582119869(119909 119910) where

119863 (119880119882) = maxsup119906isin119880

119869 (119906119882) sup119908isin119882

119869 (119880 119908)

119880119882 isin 2119883

(76)



119864 and sup119906isin119880

inf119908isin119880

119869(119906 119908) = sup119906isin119880

119869(119906 119906) = 119901(119906 119906) =

0 = 0 Thus 4119863(119879(119909) 119879(119910)) = 0 le 120582119869(119909 119910)



inf119908isin119882

119869(119906 119908) =

sup119906isin119880

inf119908isin119882

(119908 minus 119906)4 = sup

119906isin119880(4 minus 119906)

4= 81 and

sup119908isin119882

inf119906isin119880

119869(119906 119908) = sup119908isin119882

inf119906isin119880

(119908 minus 119906)4 =

sup119908isin119882





119869(119906 119908) = sup119906isin119880

inf119908isin119882

119901(119906 119908) =

sup119908isin119882

inf119906isin119880

119869(119906 119908) = sup119908isin119882

inf119906isin119880

119901(119906 119908) = 0Therefore 8119863(119879(119909) 119879(119910)) = 0 le 120582119869(119909 119910)




exist119910isin119879(119909)

119869(119909




120573isin(0infin)119869(1199090

1199100) lt 119869(1199090 119879(1199090)) + 120573



120573isin(0infin)119869(1199090 1199100) lt 119869(1199090 119879(1199090)) + 120573


120573isin(0infin)119869(1199090 1199100) lt

119869(1199090 119879(1199090)) + 120573



119869(1199090 1199100) lt 119869(1199090 119879(1199090)) +

120573


120573isin(0infin)119869(1199090 1199100) lt 119869(1199090 119879(1199090)) + 120573


1199080isin 119883 In fact if1199080

isin 119883 and (119908119898


119898+1isin 119879(119908

119898


sup119899gt119898

119869(119908119898

119908119899

) =


sup119899gt119898

119869(119908119899

119908119898

) = 0) then forall119898ge2 119908

119898



119901(119908119908119898

) =


119898rarrinfin119901(119908

119898



119869(119908 119908119898

) =


119898rarrinfin119869(119908

119898





(119909119898 119898isinN)= (LIM119877minusP81

(119909119898 119898isinN)= ))


119898 isin N) and (119906119898

119898 isin N)


119898isin 119879(119906

119898) Then forall

119898ge2 119909119898

isin [1 2] 2 isin





following


0isin 119883




0isin 119883





119909119910isin1198838119863(119879(119909)


119901(119906119882) sup



119909119910isin1198838119863(119879(119909) 119879(119910)) le 1205820119901(119909 119910) Letting 1199090 = 2


[1 2] = 119880 119879(1199100) = (4 6) = 119882 sup119906isin[12]119901(119906 (4 6)) =

sup119906isin[12](4 minus 119906)

4= 34 = 81 and sup

119908isin(46)119901([1 2] 119908) =




119862A)


119862A whereJ



119862A)whenJ119862A = P


119862A





119860 = 1198601 cup1198602 1198601 = (0 2] 1198602 = [4 6) (77)

Let 119901 1198832


119901 (119906 119908) =

0 if 119860 cap 119906 119908 = 119906 119908

120574 if 119860 cap 119906 119908 = 119906 119908

(78)

and letJ11 = P


119879 (119906) =

1198602 for 119906 isin (0 3)

119860 for 119906 = 3

1198601 for 119906 isin (3 6)

(79)


Section 11(2) (119883 119879) is a (D



119909119910isin119883119863

119871minusP1112119883 (119879(119909) 119879(119910)) le 120582119901(119909


and maxsup119906isin119879(119909)

119901(119906 119879(119910)) sup119908isin119879(119910)



exist119910isin119879(119909)



119901 (1199090 1199100) = 119901 (1199090 119879 (1199090))

=

0 if 1199090 isin 1198601

120574 for 1199090 isin (0 3) 1198601

(80)

Therefore 119901(1199090 1199100) lt 119901(1199090 119879(1199090)) + 120573



119901 (1199090 1199100) = 119901 (1199090 119879 (1199090))

=

0 if 1199090 isin 1198602

120574 for 1199090 isin (3 6) 1198602

(81)







sup119899gt119898

119901(119908119898

119908119899


lim119898rarrinfin

119901(119908119908119898


(79)forall119898⩾1 119908

119898

isin 119879(119908119898minus1



(119908119898119898isin0cupN)




119898 isin N) sub 119879[2]

(119883) = 119860



119898 isin N) and (119906119898


119898isin 119879(119906


exist1198980isinN

forall119898⩾1198980

119909119898



and Fix(119879[2]) = 119860

(6) For (119883P11)P11 = 119901 J

11 = P11 and



isin

119883 1199081isin 119879(119908

0) and 119908

2= 119906 isin 119879(119908


and forall119898⩾3 119908

119898

= 119906 then sequence (119908119898




1198832


119901 (119906 119908) =

0 if 119906 ge 119908

(119906 minus 119908)2 if 119906 lt 119908

(82)


119879 (119906) = ([0 3) cup (3 6]) 119906 for 119906 isin [0 6] (83)

Let

119864 = [0 3) cup (3 6] (84)


2rarr [0infin) is

of the form

119869 (119906 119908) =

119901 (119906 119908) if 119864 cap 119906 119908 = 119906 119908

120583 if 119864 cap 119906 119908 = 119906 119908

(85)


119869(0 6) = 36 and 119869(6 0) = 0(2)J

21 = 119869 isin J119871(119883P21)

capJ119877(119883P21)

SeeTheorem 14




[0 1) and

119863(119880119882) = maxsup119906isin119880

119869 (119906119882) sup119908isin119882

119869 (119880 119908)

119880119882 isin 2119883

(86)




119906

cup 119882119906and

inf119908isin119882

119869 (119906 119908)

le

inf119908isin119882119906119902 (119906 119908) = 0 if 119882

119906

= 119908 isin 119882 119906 ge 119908 =

inf119908isin119882119906

(119906 minus 119908)2= 0 if 119882

119906= 119908 isin 119882 119906 lt 119908 =

(87)


cup 119880119908and

inf119906isin119880

119869 (119906 119908)

le

inf119906isin119880119908119902 (119906 119908) = 0 if 119880

119906

= 119906 isin 119880 119906 ge 119908 =

inf119906isin119880119908

(119906 minus 119908)2= 0 if 119880

119906= 119906 isin 119880 119906 lt 119908 =

(88)

By (86) 2119863(119879(119909) 119879(119910)) = 0 le 120582119869(119909 119910)

Case 2 If 119909 = 119910 = 3 then 119869(119909 119910) = 120583 and 119879(119909) =


2119863(119880119880) = 0 le 120582119869(119909 119910)



inf119908isin119882

119869(119906 119908) =


119869(119906 119908) =


inf119906isin119880


119908

cup 119880119908and

inf119906isin119880

119869 (119906 119908)

le

inf119906isin119880119908119902 (119906 119908) = 0 if 119880

119906

= 119906 isin 119880 119906 ge 119908 =

inf119906isin119880119908

(119906 minus 119908)2= 0 if 119880

119906= 119906 isin 119880 119906 lt 119908 =

(89)

Thus 2119863(119879(119909) 119879(119910)) = 0 ⩽ 120582119869(119909 119910)


(3 6]) 119910 = 119882 sub 119864 and sup119906isin119880

inf119908isin119882

119869(119906 119908) = 0 sincefor 119906 isin 119880

inf119908isin119882

119869 (119906 119908)

le

inf119908isin119882119906119902 (119906 119908) = 0 if 119882

119906

= 119908 isin 119882 119906 ge 119908 =

inf119908isin119882119906

(119906 minus 119908)2= 0 if 119882

119906= 119908 isin 119882 119906 lt 119908 =

(90)


inf119906isin119880

119869(119906 119908) = 0 since inf119906isin119880

119869(119906 119908) = 119869(119908119908) = 0 for 119908 isin 119882 Thus 2119863(119879(119909) 119879(119910)) = 0 le 120582119869(119909 119910)



exist119910isin119879(119909)

119869(119909




inf119908isin119882

119869(1199090 119908) = 0 since

inf119908isin119882

119869 (1199090 119908)

le

inf119908isin1198821199090119902 (1199090 119908) = 0 if 119882

1199090 = 119908 isin 119882 1199090 ge 119908 =

inf119908isin1198821199090

(1199090 minus 119908)2= 0 if 119882

1199090= 119908 isin 119882 1199090 lt 119908 =

(91)



12)

then 119869(1199090 1199100) lt 119869(1199090 119879(1199090)) + 120574



119869(1199090 1199100) lt 119869(1199090 119879(1199090)) + 120574




119869(1199090 119879(1199090)) + 120574


1199080


119898



sup119899gt119898

119869(119908119898

119908119899


lim119898rarrinfin

119869(119908 119908119898



1isin 119879(119908

0) = ([0 3) cup

(3 6]) 1199080 and forall

119898ge2 119908119898

isin 119879(119908119898minus1


(119908119898119898isin0cupN)

Case 2 If 1199080= 3 then 119908

1isin 119879(119908

0) = [0 3) cup (3 6] 1199082

isin

119879(1199081) = ([0 3) cup (3 6]) 119908

1 and forall

119898⩾3 119908119898

isin 119879(119908119898minus1

) sub




0isin 119883 and



119898rarrinfinsup

119899gt119898119869(119908

119898

119908119899

) =

0 of (119908119898


[2]) is a left P


119898 119898 isin N) sub 119879

[2](119883) =



119898 isin N) and (119906119898

119898 isin N)


119898isin 119879(119906


exist1198980isinN

forall119898ge1198980

119909119898


(119909119898 119898isinN)and 6 isin 119879

[2](6)

(7) ForP21 = 119901J



that 6 isin 119879[2]


0isin 119883






11 = 119901


119879 (119906) =

4 for 119906 isin (0 3)

2 for 119906 isin [3 6) (92)



119909119910isin119883119863

119871minusP111119883 (119879(119909) 119879(119910)) =

max119901(119879(119909) 119879(119910)) 119901(119879(119910) 119879(119909)) le 120582119901(119909 119910) and Fix(119879) =






119898+1= 119879(119908

119898


sup119899gt119898

119901(119908119898

119908119899

) =


119898



(119908119898119898isin0cupN)




119898 isin N) sub

119879[2]



119898 isin N) and (119906119898

119898 isin N)


119898= 119879

[2](119906

119898) then by (77) (78) and (92)



(2) isin

LIM119871minusP11(119909119898 119898isinN)

and 4 = 119879[2]





119883119860 then 119901(119906 119908) = 119901(119908 119906) = 120574 gt 0(6) For P

11 = 119901 (119883 119879) and J11 = P





119889


119889



(119883 119889) rarr (CB(119883)119867119889











References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119860 = 1198601 cup1198602 1198601 = (0 2] 1198602 = [4 6) (77)

Let 119901 1198832


119901 (119906 119908) =

0 if 119860 cap 119906 119908 = 119906 119908

120574 if 119860 cap 119906 119908 = 119906 119908

(78)

and letJ11 = P


119879 (119906) =

1198602 for 119906 isin (0 3)

119860 for 119906 = 3

1198601 for 119906 isin (3 6)

(79)


Section 11(2) (119883 119879) is a (D



119909119910isin119883119863

119871minusP1112119883 (119879(119909) 119879(119910)) le 120582119901(119909


and maxsup119906isin119879(119909)

119901(119906 119879(119910)) sup119908isin119879(119910)



exist119910isin119879(119909)



119901 (1199090 1199100) = 119901 (1199090 119879 (1199090))

=

0 if 1199090 isin 1198601

120574 for 1199090 isin (0 3) 1198601

(80)

Therefore 119901(1199090 1199100) lt 119901(1199090 119879(1199090)) + 120573



119901 (1199090 1199100) = 119901 (1199090 119879 (1199090))

=

0 if 1199090 isin 1198602

120574 for 1199090 isin (3 6) 1198602

(81)







sup119899gt119898

119901(119908119898

119908119899


lim119898rarrinfin

119901(119908119908119898


(79)forall119898⩾1 119908

119898

isin 119879(119908119898minus1



(119908119898119898isin0cupN)




119898 isin N) sub 119879[2]

(119883) = 119860



119898 isin N) and (119906119898


119898isin 119879(119906


exist1198980isinN

forall119898⩾1198980

119909119898



and Fix(119879[2]) = 119860

(6) For (119883P11)P11 = 119901 J

11 = P11 and



isin

119883 1199081isin 119879(119908

0) and 119908

2= 119906 isin 119879(119908


and forall119898⩾3 119908

119898

= 119906 then sequence (119908119898




1198832


119901 (119906 119908) =

0 if 119906 ge 119908

(119906 minus 119908)2 if 119906 lt 119908

(82)


119879 (119906) = ([0 3) cup (3 6]) 119906 for 119906 isin [0 6] (83)

Let

119864 = [0 3) cup (3 6] (84)


2rarr [0infin) is

of the form

119869 (119906 119908) =

119901 (119906 119908) if 119864 cap 119906 119908 = 119906 119908

120583 if 119864 cap 119906 119908 = 119906 119908

(85)


119869(0 6) = 36 and 119869(6 0) = 0(2)J

21 = 119869 isin J119871(119883P21)

capJ119877(119883P21)

SeeTheorem 14




[0 1) and

119863(119880119882) = maxsup119906isin119880

119869 (119906119882) sup119908isin119882

119869 (119880 119908)

119880119882 isin 2119883

(86)




119906

cup 119882119906and

inf119908isin119882

119869 (119906 119908)

le

inf119908isin119882119906119902 (119906 119908) = 0 if 119882

119906

= 119908 isin 119882 119906 ge 119908 =

inf119908isin119882119906

(119906 minus 119908)2= 0 if 119882

119906= 119908 isin 119882 119906 lt 119908 =

(87)


cup 119880119908and

inf119906isin119880

119869 (119906 119908)

le

inf119906isin119880119908119902 (119906 119908) = 0 if 119880

119906

= 119906 isin 119880 119906 ge 119908 =

inf119906isin119880119908

(119906 minus 119908)2= 0 if 119880

119906= 119906 isin 119880 119906 lt 119908 =

(88)

By (86) 2119863(119879(119909) 119879(119910)) = 0 le 120582119869(119909 119910)

Case 2 If 119909 = 119910 = 3 then 119869(119909 119910) = 120583 and 119879(119909) =


2119863(119880119880) = 0 le 120582119869(119909 119910)



inf119908isin119882

119869(119906 119908) =


119869(119906 119908) =


inf119906isin119880


119908

cup 119880119908and

inf119906isin119880

119869 (119906 119908)

le

inf119906isin119880119908119902 (119906 119908) = 0 if 119880

119906

= 119906 isin 119880 119906 ge 119908 =

inf119906isin119880119908

(119906 minus 119908)2= 0 if 119880

119906= 119906 isin 119880 119906 lt 119908 =

(89)

Thus 2119863(119879(119909) 119879(119910)) = 0 ⩽ 120582119869(119909 119910)


(3 6]) 119910 = 119882 sub 119864 and sup119906isin119880

inf119908isin119882

119869(119906 119908) = 0 sincefor 119906 isin 119880

inf119908isin119882

119869 (119906 119908)

le

inf119908isin119882119906119902 (119906 119908) = 0 if 119882

119906

= 119908 isin 119882 119906 ge 119908 =

inf119908isin119882119906

(119906 minus 119908)2= 0 if 119882

119906= 119908 isin 119882 119906 lt 119908 =

(90)


inf119906isin119880

119869(119906 119908) = 0 since inf119906isin119880

119869(119906 119908) = 119869(119908119908) = 0 for 119908 isin 119882 Thus 2119863(119879(119909) 119879(119910)) = 0 le 120582119869(119909 119910)



exist119910isin119879(119909)

119869(119909




inf119908isin119882

119869(1199090 119908) = 0 since

inf119908isin119882

119869 (1199090 119908)

le

inf119908isin1198821199090119902 (1199090 119908) = 0 if 119882

1199090 = 119908 isin 119882 1199090 ge 119908 =

inf119908isin1198821199090

(1199090 minus 119908)2= 0 if 119882

1199090= 119908 isin 119882 1199090 lt 119908 =

(91)



12)

then 119869(1199090 1199100) lt 119869(1199090 119879(1199090)) + 120574



119869(1199090 1199100) lt 119869(1199090 119879(1199090)) + 120574




119869(1199090 119879(1199090)) + 120574


1199080


119898



sup119899gt119898

119869(119908119898

119908119899


lim119898rarrinfin

119869(119908 119908119898



1isin 119879(119908

0) = ([0 3) cup

(3 6]) 1199080 and forall

119898ge2 119908119898

isin 119879(119908119898minus1


(119908119898119898isin0cupN)

Case 2 If 1199080= 3 then 119908

1isin 119879(119908

0) = [0 3) cup (3 6] 1199082

isin

119879(1199081) = ([0 3) cup (3 6]) 119908

1 and forall

119898⩾3 119908119898

isin 119879(119908119898minus1

) sub




0isin 119883 and



119898rarrinfinsup

119899gt119898119869(119908

119898

119908119899

) =

0 of (119908119898


[2]) is a left P


119898 119898 isin N) sub 119879

[2](119883) =



119898 isin N) and (119906119898

119898 isin N)


119898isin 119879(119906


exist1198980isinN

forall119898ge1198980

119909119898


(119909119898 119898isinN)and 6 isin 119879

[2](6)

(7) ForP21 = 119901J



that 6 isin 119879[2]


0isin 119883






11 = 119901


119879 (119906) =

4 for 119906 isin (0 3)

2 for 119906 isin [3 6) (92)



119909119910isin119883119863

119871minusP111119883 (119879(119909) 119879(119910)) =

max119901(119879(119909) 119879(119910)) 119901(119879(119910) 119879(119909)) le 120582119901(119909 119910) and Fix(119879) =






119898+1= 119879(119908

119898


sup119899gt119898

119901(119908119898

119908119899

) =


119898



(119908119898119898isin0cupN)




119898 isin N) sub

119879[2]



119898 isin N) and (119906119898

119898 isin N)


119898= 119879

[2](119906

119898) then by (77) (78) and (92)



(2) isin

LIM119871minusP11(119909119898 119898isinN)

and 4 = 119879[2]





119883119860 then 119901(119906 119908) = 119901(119908 119906) = 120574 gt 0(6) For P

11 = 119901 (119883 119879) and J11 = P





119889


119889



(119883 119889) rarr (CB(119883)119867119889











References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119906

cup 119882119906and

inf119908isin119882

119869 (119906 119908)

le

inf119908isin119882119906119902 (119906 119908) = 0 if 119882

119906

= 119908 isin 119882 119906 ge 119908 =

inf119908isin119882119906

(119906 minus 119908)2= 0 if 119882

119906= 119908 isin 119882 119906 lt 119908 =

(87)


cup 119880119908and

inf119906isin119880

119869 (119906 119908)

le

inf119906isin119880119908119902 (119906 119908) = 0 if 119880

119906

= 119906 isin 119880 119906 ge 119908 =

inf119906isin119880119908

(119906 minus 119908)2= 0 if 119880

119906= 119906 isin 119880 119906 lt 119908 =

(88)

By (86) 2119863(119879(119909) 119879(119910)) = 0 le 120582119869(119909 119910)

Case 2 If 119909 = 119910 = 3 then 119869(119909 119910) = 120583 and 119879(119909) =


2119863(119880119880) = 0 le 120582119869(119909 119910)



inf119908isin119882

119869(119906 119908) =


119869(119906 119908) =


inf119906isin119880


119908

cup 119880119908and

inf119906isin119880

119869 (119906 119908)

le

inf119906isin119880119908119902 (119906 119908) = 0 if 119880

119906

= 119906 isin 119880 119906 ge 119908 =

inf119906isin119880119908

(119906 minus 119908)2= 0 if 119880

119906= 119906 isin 119880 119906 lt 119908 =

(89)

Thus 2119863(119879(119909) 119879(119910)) = 0 ⩽ 120582119869(119909 119910)


(3 6]) 119910 = 119882 sub 119864 and sup119906isin119880

inf119908isin119882

119869(119906 119908) = 0 sincefor 119906 isin 119880

inf119908isin119882

119869 (119906 119908)

le

inf119908isin119882119906119902 (119906 119908) = 0 if 119882

119906

= 119908 isin 119882 119906 ge 119908 =

inf119908isin119882119906

(119906 minus 119908)2= 0 if 119882

119906= 119908 isin 119882 119906 lt 119908 =

(90)


inf119906isin119880

119869(119906 119908) = 0 since inf119906isin119880

119869(119906 119908) = 119869(119908119908) = 0 for 119908 isin 119882 Thus 2119863(119879(119909) 119879(119910)) = 0 le 120582119869(119909 119910)



exist119910isin119879(119909)

119869(119909




inf119908isin119882

119869(1199090 119908) = 0 since

inf119908isin119882

119869 (1199090 119908)

le

inf119908isin1198821199090119902 (1199090 119908) = 0 if 119882

1199090 = 119908 isin 119882 1199090 ge 119908 =

inf119908isin1198821199090

(1199090 minus 119908)2= 0 if 119882

1199090= 119908 isin 119882 1199090 lt 119908 =

(91)



12)

then 119869(1199090 1199100) lt 119869(1199090 119879(1199090)) + 120574



119869(1199090 1199100) lt 119869(1199090 119879(1199090)) + 120574




119869(1199090 119879(1199090)) + 120574


1199080


119898



sup119899gt119898

119869(119908119898

119908119899


lim119898rarrinfin

119869(119908 119908119898



1isin 119879(119908

0) = ([0 3) cup

(3 6]) 1199080 and forall

119898ge2 119908119898

isin 119879(119908119898minus1


(119908119898119898isin0cupN)

Case 2 If 1199080= 3 then 119908

1isin 119879(119908

0) = [0 3) cup (3 6] 1199082

isin

119879(1199081) = ([0 3) cup (3 6]) 119908

1 and forall

119898⩾3 119908119898

isin 119879(119908119898minus1

) sub




0isin 119883 and



119898rarrinfinsup

119899gt119898119869(119908

119898

119908119899

) =

0 of (119908119898


[2]) is a left P


119898 119898 isin N) sub 119879

[2](119883) =



119898 isin N) and (119906119898

119898 isin N)


119898isin 119879(119906


exist1198980isinN

forall119898ge1198980

119909119898


(119909119898 119898isinN)and 6 isin 119879

[2](6)

(7) ForP21 = 119901J



that 6 isin 119879[2]


0isin 119883






11 = 119901


119879 (119906) =

4 for 119906 isin (0 3)

2 for 119906 isin [3 6) (92)



119909119910isin119883119863

119871minusP111119883 (119879(119909) 119879(119910)) =

max119901(119879(119909) 119879(119910)) 119901(119879(119910) 119879(119909)) le 120582119901(119909 119910) and Fix(119879) =






119898+1= 119879(119908

119898


sup119899gt119898

119901(119908119898

119908119899

) =


119898



(119908119898119898isin0cupN)




119898 isin N) sub

119879[2]



119898 isin N) and (119906119898

119898 isin N)


119898= 119879

[2](119906

119898) then by (77) (78) and (92)



(2) isin

LIM119871minusP11(119909119898 119898isinN)

and 4 = 119879[2]





119883119860 then 119901(119906 119908) = 119901(119908 119906) = 120574 gt 0(6) For P

11 = 119901 (119883 119879) and J11 = P





119889


119889



(119883 119889) rarr (CB(119883)119867119889











References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














11 = 119901


119879 (119906) =

4 for 119906 isin (0 3)

2 for 119906 isin [3 6) (92)



119909119910isin119883119863

119871minusP111119883 (119879(119909) 119879(119910)) =

max119901(119879(119909) 119879(119910)) 119901(119879(119910) 119879(119909)) le 120582119901(119909 119910) and Fix(119879) =






119898+1= 119879(119908

119898


sup119899gt119898

119901(119908119898

119908119899

) =


119898



(119908119898119898isin0cupN)




119898 isin N) sub

119879[2]



119898 isin N) and (119906119898

119898 isin N)


119898= 119879

[2](119906

119898) then by (77) (78) and (92)



(2) isin

LIM119871minusP11(119909119898 119898isinN)

and 4 = 119879[2]





119883119860 then 119901(119906 119908) = 119901(119908 119906) = 120574 gt 0(6) For P

11 = 119901 (119883 119879) and J11 = P





119889


119889



(119883 119889) rarr (CB(119883)119867119889











References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Date post:	25-Feb-2021
Category:	Documents
Upload:	others
View:	0 times
Download:	0 times

Research Article Quasi-Triangular Spaces, Pompeiu...

Documents