Research Article Application of Coupled Fixed Point...

Research ArticleApplication of Coupled Fixed Point Technique inSolving Integral Equations on Modified IntuitionisticFuzzy Metric Spaces

Bhavana Deshpande and Amrish Handa

Department of Mathematics Govt PG Arts and Science College Ratlam 457001 India

Correspondence should be addressed to Bhavana Deshpande bhavnadeshpandeyahoocom

Received 7 January 2014 Accepted 18 May 2014 Published 22 June 2014

Academic Editor RustomM Mamlook

Copyright copy 2014 B Deshpande and A HandaThis 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

We establish a common coupled fixed point theorem for weakly compatible mappings on modified intuitionistic fuzzy metricspaces As an application of our result we study the existence and uniqueness of the solution to a nonlinear Fredholm integralequation We also give an example to demonstrate our result

1 Introduction

The concept of fuzzy metric space has been introduced inseveral ways In [1] Kramosil and Michalek introduced theconcept of fuzzy metric space Later on it is modified byGeorge and Veeramani [2] with the help of continuous t-norms and they defined the Hausdorff topology of fuzzymetric spaces

Atanassov [3] introduced and studied the concept ofintuitionistic fuzzy sets as a generalization of fuzzy setsAlaca et al [4] using the idea of intuitionistic fuzzy setsdefined the notion of intuitionistic fuzzy metric space withthe help of continuous t-norms and continuous t-conormsas a generalization of fuzzy metric space due to Kramosiland Michalek [1] In [5] Park generalized the notion offuzzy metric space given by George and Veeramani [2] andintroduced the notion of intuitionistic fuzzy metric space

Gregori et al [6] pointed out that topologies generated byfuzzy metric and intuitionistic fuzzy metric coincide In viewof this observation Saadati et al [7] modified the notion ofintuitionistic fuzzymetric and defined the notion ofmodifiedintuitionistic fuzzy metric spaces with the help of continuoust-representable

Bhaskar and Lakshmikantham [8] introduced the notionof coupled fixed point and mixed monotone mappings and

gave some coupled fixed point theorems As an applicationthey study the existence and uniqueness of solution for peri-odic boundary value problems Lakshmikantham and Ciric[9] introduced the concept of coupled coincidence point andproved some common coupled fixed point theorems Sedghiet al [10] gave a coupled fixed point theorem for contractionsin fuzzy metric space which was further generalized by Hu[11] In [12] Hu et al improved rectified and generalized theresult obtained in [11]

On the other hand many scientific and engineeringproblems can be described by integral equations Initial andboundary value problems can be transformed into Volterraor Fredholm integral equations Integral equations can alsobe created by many mathematical physics models suchas diffraction problems scattering in quantum mechanicsconformal mapping and water wave Integral equationsor integro-differential equations can be applied in scienceand engineering Many areas that are described by integralequations are Volterrarsquos population growth model biologicalspecies living together propagation of stocked fish in a newlake the heat radiation and so forth

Very recently Deshpande et al [13] proved a commonfixed point theorem formappings under 120601-contractive condi-tions on intuitionistic fuzzy metric spaces As an application

Hindawi Publishing CorporationAdvances in Fuzzy SystemsVolume 2014 Article ID 348069 11 pageshttpdxdoiorg1011552014348069

2 Advances in Fuzzy Systems

they study the existence and uniqueness of the solution to anonlinear Fredholm integral equation

In this paper we prove a common coupled fixed pointtheorem for weakly compatible mappings on modifiedintuitionistic fuzzy metric spaces As an application of ourresult we study the existence and uniqueness of the solutionto a nonlinear Fredholm integral equation which arisenaturally in the theory of signal processing linear forwardmodeling and inverse problems We also give an example tovalidate our result We extend and generalize the results ofHu [11] Hu et al [12] and Sedghi et al [10] in the settings ofmodified intuitionistic fuzzy metric spaces The result is thegenuine generalization of the result of Deshpande et al [13]

2 Preliminaries

Lemma 1 (Deschrijver and Kerre [14]) Consider the set 119871lowastand operation le

119871lowastdefined by

119871lowast

= (1199091 1199092) (1199091 1199092) isin [0 1]

2

1199091+ 1199092le 1 (1)

(1199091 1199092)le119871lowast(1199101 1199102) hArr 119909

1le 1199101 and 119909

2ge 1199102for every

(1199091 1199092) (1199101 1199102) isin 119871lowast Then (119871lowast le

119871lowast) is a complete lattice

Definition 2 (Atanassov [3]) An intuitionistic fuzzy setA120577120578

in a universe 119880 is an object A120577120578

= 120577A(119906) 120578A(119906) wherefor all 119906 isin 119880 120577A(119906) isin [0 1] and 120578A(119906) isin [0 1] arecalled the membership degree and nonmembership degreerespectively of 119906 in A

120577120578and further they satisfy 120577A(119906) +

120578A(119906) le 1 For every 119911119894= (119909119894 119910119894) isin 119871lowast if 119888

119894isin [0 1] such

that sum119899119895=1

119888119895= 1 then it is easy to see that

1198881(1199091 1199101) + sdot sdot sdot + 119888

119899(119909119899 119910119899)

=119899

sum119895=1

119888119895(119909119895 119910119895) = (

119899

sum119895=1

119888119895119909119895119899

sum119895=1

119888119895119910119895) isin 119871

lowast

(2)

We denote its units by 0119871lowast = (0 1) and 1

119871lowast = (1 0)

Classically a triangular norm lowast = 119879 on [0 1] is defined as anincreasing commutative associative mapping 119879 [0 1]2 rarr[0 1] satisfying 119879(1 119909) = 1 lowast 119909 = 119909 for all 119909 isin [0 1]A triangular conorm 119878 = is defined as an increasingcommutative associative mapping 119878 [0 1]2 rarr [0 1]satisfying 119878(0 119909) = 0 119909 = 119909 for all 119909 isin [0 1] Usingthe lattice (119871lowast le

119871lowast) these definitions can be straightforwardly

extended

Definition 3 (Deschrijver et al [15]) A triangular norm (t-norm) on 119871lowast is a mapping T (119871lowast)2 rarr 119871lowast satisfying thefollowing conditions

(forall119909 isin 119871lowast)(T(119909 1119871lowast) = 119909) (boundary condition)

(forall(119909 119910) isin (119871lowast)2

) (T(119909 119910) = T(119910 119909)) (commutativ-ity)(forall(119909 119910 119911) isin (119871lowast)

3

) (T(119909T(119910 119911)) = T(T(119909 119910) 119911))(associativity)(forall(119909 1199091015840 119910 1199101015840) isin (119871lowast)

4

) (119909le119871lowast1199091015840 and 119910le

119871lowast1199101015840 rArr

T(119909 119910)le119871lowastT(1199091015840 1199101015840)) (monotonicity)

Definition 4 (Deschrijver and Kerre [14] and Deschrijver etal [15]) A continuous t-normT on 119871lowast is called continuoust-representable if and only if there exist a continuous t-normlowast and a continuous t-conorm on [0 1] such that for all119909 = (119909

1 1199092) 119910 = (119910

1 1199102) isin 119871lowast

T (119909 119910) = (1199091lowast 1199101 1199092 1199102) (3)

Now define a sequenceT119899 recursively byT1 = T and

T119899

(119909(1)

119909(119899+1)

) = T (T119899minus1

(119909(1)

119909(119899)

) 119909(119899+1)

)

(4)

for 119899 ge 2 and 119909(119894) isin 119871lowast

Definition 5 (Deschrijver and Kerre [14] andDeschrijver et al[15]) A negator on 119871lowast is any decreasing mappingN 119871lowast rarr119871lowast satisfyingN(0

119871lowast) = 1

119871lowast andN(1

119871lowast) = 0

119871lowast IfN(N(119909)) =

119909 for all 119909 isin 119871lowast then N is called an involutive negator Anegator on [0 1] is a decreasing mapping119873 [0 1] rarr [0 1]satisfying 119873(0) = 1 and 119873(1) = 0 119873

119904denotes the standard

negator on [0 1] defined as for all 119909 isin [0 1]119873119904(119909) = 1 minus 119909

Definition 6 (Saadati et al [7]) Let119872119873 be fuzzy sets from1198832 times (0 + infin) to [0 1] such that 119872(119909 119910 119905) + 119873(119909 119910 119905) le 1for all 119909 119910 isin 119883 and 119905 gt 0 The 3-tuple (119883 M

119872119873T) is said

to be a modified intuitionistic fuzzy metric space if 119883 is anarbitrary (nonempty) setT is a continuous t-representableandM

119872119873is a mapping 1198832 times (0 + infin) rarr 119871lowast satisfying the

following conditions for every 119909 119910 119911 isin 119883 and 119905 119904 gt 0

(a) M119872119873

(119909 119910 119905)gt119871lowast0119871lowast

(b) M119872119873

(119909 119910 119905) = 1119871lowast if and only if 119909 = 119910

(c) M119872119873

(119909 119910 119905) = M119872119873

(119910 119909 119905)(d) M

119872119873(119909 119910 119905+119904)ge

119871lowastT(M

119872119873(119909 119911 119905)M

119872119873(119911 119910 119904))

(e) M119872119873

(119909 119910 sdot) (0 infin) rarr 119871lowast is continuous

In this caseM119872119873

is called amodified intuitionistic fuzzymetric Here

M119872119873

(119909 119910 119905) = (119872(119909 119910 119905)119873(119909 119910 119905))

Example 7 (Saadati et al [7]) Let (119883 119889) be a metric spaceDenoteT(119886 119887) = (119886

11198871min119886

2+1198872 1) for all 119886 = (119886

1 1198862) and

119887 = (1198871 1198872) isin 119871lowast and let119872 and119873 be fuzzy sets on1198832times(0infin)

defined as follows

M119872119873

(119909 119910 119905) = (119872 (119909 119910 119905) 119873 (119909 119910 119905))

= (ℎ119905119899

ℎ119905119899 + 119898119889 (119909 119910)

119898119889 (119909 119910)

ℎ119905119899 + 119898119889 (119909 119910))

(5)

for all 119905 ℎ 119898 119899 isin 119877+ Then (119883M119872119873

T) is a modifiedintuitionistic fuzzy metric space

Example 8 (Saadati et al [7]) Let 119883 = 119873 DenoteT(119886 119887) =(max0 119886

1+ 1198871minus 1 119886

2+ 1198872minus 11988621198872) for all 119886 = (119886

1 1198862) and

Advances in Fuzzy Systems 3


defined as follows

M119872119873

(119909 119910 119905) = (119872 (119909 119910 119905) 119873 (119909 119910 119905))

=

(119909

119910119910 minus 119909

119910) if 119909 le 119910

(119910

119909119909 minus 119910

119909) if 119910 le 119909

(6)

for all 119909 119910 isin 119883 and 119905 gt 0 Then (119883M119872119873


Lemma 9 (Fang [16]) We say that the intuitionistic fuzzymetric space (119883M

119872119873T) has property (119862) if it satisfies the

following condition M119872119873

(119909 119910 119905) = 119862 for all 119905 gt 0 implies119862 = 1

119871lowast

Remark 10 Throughout this paper (119883M119872119873

T) is a modi-fied intuitionistic fuzzy metric space with property (119862)

Definition 11 (Saadati et al [7]) A sequence 119909119899 in amodified

intuitionistic fuzzy metric space (119883M119872119873

T) is called aCauchy sequence if for each 0 lt 120576 lt 1 and 119905 gt 0 thereexists 119899

0isin 119873 such that

M119872119873

(119909119899 119909119898 119905) gt119871lowast (119873119904(120576) 120576) (7)

and for each 119899119898 ge 1198990 here 119873

119904is the standard negator

The sequence 119909119899 is said to be convergent to 119909 isin 119883 in the

modified intuitionistic fuzzy metric space (119883M119872119873

T) and

denoted by 119909119899

M119872119873

997888997888997888997888rarr 119909 if M119872119873

(119909119899 119909 119905) rarr 1

119871lowast whenever

119899 rarr infin for every 119905 gt 0 A modified intuitionistic fuzzymetric space is said to be complete if and only if every Cauchysequence is convergent

Lemma 12 (Saadati and Park [17]) Let M119872119873

be a modifiedintuitionistic fuzzy metric Then for any 119905 gt 0M

119872119873(119909 119910 119905)

is nondecreasing with respect to 119905 in (119871lowast le119871lowast) for all 119909 119910 in119883

Definition 13 (Saadati et al [7]) Let (119883M119872119873

T) be amodified intuitionistic fuzzy metric space For 119905 gt 0 definethe open ball119861(119909 119903 119905)with center119909 isin 119883 and radius 0 lt 119903 lt 1as

119861 (119909 119903 119905) = 119910 isin 119883 M119872119873

(119909 119910 119905) gt119871lowast (119873119904(119903) 119903) (8)

A subset 119860 sub 119883 is called open if for each 119909 isin 119860 thereexist 119905 gt 0 and 0 lt 119903 lt 1 such that 119861(119909 119903 119905) sub 119860 Let120591119872119873

denote the family of all open subsets of119883 120591119872119873

is calledthe topology induced by modified intuitionistic fuzzy metricThis topology is Hausdorff


T) be amodified intuitionistic fuzzy metric space M is said to becontinuous on119883 times 119883 times (0infin) if

lim119899rarrinfin

M119872119873

(119909119899 119910119899 119905119899) = M

119872119873(119909 119910 119905) (9)

whenever a sequence (119909119899 119910119899 119905119899) in119883times119883times(0infin) converges

to a point (119909 119910 119905) isin 119883 times 119883 times (0infin) that is

lim119899rarrinfin

M119872119873

(119909119899 119910 119905) = M

119872119873(119909 119910 119905)

lim119899rarrinfin

M119872119873

(119909 119910119899 119905) = M

119872119873(119909 119910 119905)

lim119899rarrinfin

M119872119873

(119909 119910 119905119899) = M

119872119873(119909 119910 119905)

(10)

Lemma 15 (Saadati et al [7]) Let (119883M119872119873

T) be a mod-ified intuitionistic fuzzy metric space Then M is continuousfunction on 119883 times 119883 times (0infin)

Definition 16 (Bhaskar and Lakshmikantham [8]) An ele-ment (119909 119910) isin 119883 times 119883 is called a coupled fixed point of themapping 119865 119883 times 119883 rarr 119883 if

119865 (119909 119910) = 119909 119865 (119910 119909) = 119910 (11)

Definition 17 (Lakshmikantham and Ciric [9]) Let 119883 be anonempty set An element (119909 119910) isin 119883 times 119883 is called a coupledcoincidence point of the mappings 119865 119883 times 119883 rarr 119883 and119892 119883 rarr 119883 if

119865 (119909 119910) = 119892 (119909) 119865 (119910 119909) = 119892 (119910) (12)

Definition 18 (Lakshmikantham and Ciric [9]) Let 119883 be anonempty set An element (119909 119910) isin 119883times119883 is called a commoncoupled fixed point of the mappings 119865 119883 times 119883 rarr 119883 and119892 119883 rarr 119883 if

119909 = 119865 (119909 119910) = 119892 (119909) 119910 = 119865 (119910 119909) = 119892 (119910) (13)

Definition 19 (Lakshmikantham and Ciric [9]) Let 119883 be anonempty set An element 119909 isin 119883 is called a common fixedpoint of the mappings 119865 119883 times 119883 rarr 119883 and 119892 119883 rarr 119883 if

119909 = 119865 (119909 119909) = 119892 (119909) (14)

Definition 20 (Lakshmikantham and Ciric [9]) Let 119883 be anonempty setThemappings119865 119883times119883 rarr 119883 and119892 119883 rarr 119883are said to be commutative if

119892119865 (119909 119910) = 119865 (119892119909 119892119910) forall (119909 119910) isin 119883 times 119883 (15)

Definition 21 (Fang [18]) Let (119883M119872119873

T) be a modifiedintuitionistic fuzzy metric spaceThemappings 119865 119883times119883 rarr119883 and 119892 119883 rarr 119883 are said to be compatible if

lim119899rarrinfin

M119872119873

(119892119865 (119909119899 119910119899) 119865 (119892119909

119899 119892119910119899) 119905) = 1

119871lowast

lim119899rarrinfin

M119872119873

(119892119865 (119910119899 119909119899) 119865 (119892119910

119899 119892119909119899) 119905) = 1

119871lowast

(16)

for all 119905 gt 0 whenever 119909119899 and 119910

119899 are sequences in119883 such

thatlim119899rarrinfin

119865 (119909119899 119910119899) = lim119899rarrinfin

119892 (119909119899) = 119909

lim119899rarrinfin

119865 (119910119899 119909119899) = lim119899rarrinfin

119892 (119910119899) = 119910 forall119909 119910 isin 119883

(17)

Definition 22 (Abbas et al [19]) Let 119883 be a nonempty setThe mappings 119865 119883 times 119883 rarr 119883 and 119892 119883 rarr 119883 are calledweakly compatible mappings if 119865(119909 119910) = 119892(119909) 119865(119910 119909) =119892(119910) implies that 119892119865(119909 119910) = 119865(119892119909 119892119910) and 119892119865(119910 119909) =119865(119892119910 119892119909) for all 119909 119910 isin 119883


3 Main Results

Definition 23 Let sup0lt119905lt1

T(119905 119905) = 1119871lowast A continuous t-

representableT is said to be continuous t-representable ofH-type if the family of functions T119898(119905)infin

119898=1is equicontinuous

at 119905 = 1119871lowast where

T1

(119905) = T (119905 119905) T119898+1

(119905) = T (119905T119898

(119905))

119898 = 1 2 119905 isin 119871lowast

(18)

ObviouslyT is aH-type t-representable if and only if for any0 lt 120582 lt 1 there exists 0 lt 120583 lt 1 such that

T119898

(119905) gt119871lowast (119873119904(120582) 120582) forall119898 isin 119873 when 119905gt

119871lowast (119873119904(120583) 120583)

(19)

Remark 24 In a modified intuitionistic fuzzy metric space(119883M

119872119873T) whenever M

119872119873(119909 119910 119905)gt

119871lowast(119873119904(119903) 119903) for

119909 119910 isin 119883 119905 gt 0 and 0 lt 119903 lt 1 we can find a 1199050 0 lt 119905

0lt 119905

such that M119872119873

(119909 119910 1199050) gt (119873

119904(119903) 119903)

Remark 25 For convenience we denote

[M119872119873

(119909 119910 119905)]119899

= T119899minus1

(M119872119873

(119909 119910 119905)) (20)

for all 119899 isin 119873

Definition 26 Let (119883M119872119873

T) be a modified intuitionisticfuzzy metric spaceM

119872119873is said to satisfy the p-property on

1198832 times (0infin) if

lim119901rarrinfin

[M119872119873

(119909 119910 119896119901

119905)]119901

= 1119871lowast (21)

whenever 119909 119910 isin 119883 119896 gt 1 and 119901 isin 119873

Lemma 27 Let (119883M119872119873

T) be a modified intuitionisticfuzzy metric space and letM

119872119873satisfy the p-property then

lim119905rarrinfin

M119872119873

(119909 119910 119905) = 1119871lowast forall119909 119910 isin 119883 (22)

Proof If not since M119872119873

(119909 119910 sdot) is nondecreasing thereexists 119909

0 1199100isin 119883 such that

lim119905rarrinfin

M119872119873

(1199090 1199100 119905) = (120582119873

119904(120582)) lt 1

119871lowast (23)

and then for 119896 gt 1 119896119901119905 rarr infin when 119901 rarr infin as 119905 gt 0 andwe get

lim119901rarrinfin

[M119872119873

(1199090 1199100 119896119901

119905)]119901

= 0119871lowast (24)

which is a contradiction

Remark 28 Condition (22) cannot guarantee the p-propertySee the following example

Example 29 Let (119883 119889) be an ordinary metric space and let 120595be defined as follows

120595 (119905) =

120572radic119905 0 le 119905 le 4

1 minus1

ln 119905 119905 gt 4

(25)

where 120572 = (12)(1 minus (1 ln 4)) Then 120595(119905) is continuous andincreasing in (0infin) 120595(119905) isin (0 1) and lim

119905rarrinfin120595(119905) = 1 Let

M119872119873

(119909 119910 119905) = ([120595 (119905)]119889(119909119910)

1 minus [120595 (119905)]119889(119909119910)

) (26)

for all 119909 119910 isin 119883 and 119905 gt 0 andT(119886 119887) = (11988611198871 1198862+ 1198872minus 11988621198872)

for all 119886 = (1198861 1198862) and 119887 = (119887

1 1198872) isin 119871lowast Then (119883M

119872119873T)

is a modified intuitionistic fuzzy metric space with

lim119905rarrinfin

M119872119873

(119909 119910 119905)

= lim119905rarrinfin

([120595 (119905)]119889(119909119910)

1 minus [120595 (119905)]119889(119909119910)

)

= 1119871lowast forall119909 119910 isin 119883

(27)

But for any 119909 = 119910 119896 gt 1 and 119905 gt 0

lim119901rarrinfin

[M119872119873

(119909 119910 119896119901

119905)]119901

= ( lim119901rarrinfin

[120595 (119896119901

119905)]119889(119909119910)sdot119901

lim119901rarrinfin

[1 minus [120595 (119896119901

119905)]119889(119909119910)sdot119901

])


[1 minus1

ln (119896119901119905)]119889(119909119910)sdot119901

lim119901rarrinfin

[1 minus [1 minus1

ln(119896119901119905)]119901sdot119889(119909119910)

])

= (119890minus119889(119909119910) ln 119896

1 minus 119890minus119889(119909119910) ln 119896

)

= 1119871lowast

(28)

Define Φ = 120601 119877+ rarr 119877+ where 119877+ = [0 + infin) and each120601 isin Φ satisfies the following conditions

(120601-1) 120601 is nondecreasing(120601-2) 120601 is continuous(120601-3)suminfin

119899=0120601119899(119905) lt +infin for all 119905 gt 0 where 120601119899+1(119905) =

120601119899(120601(119905)) 119899 isin 119873

It is easy to prove that if 120601 isin Φ then 120601(119905) lt 119905 for all 119905 gt 0

Lemma 30 Let (119883M119872119873

T) be a modified intuitionisticfuzzy metric space whereT is a continuous t-representable ofH-type If there exists 120601 isin Φ such that if

M119872119873

(119909 119910 120601 (119905)) ge119871lowastM119872119873

(119909 119910 119905) forall119905 gt 0 (29)

then 119909 = 119910

Proof Since 119905 gt 0 and 119905 gt 120601(119905) Using the monotonyof M119872119873

we have M119872119873

(119909 119910 119905)ge119871lowastM119872119873

(119909 119910 120601(119905)) Using(29) and the definition ofmodified intuitionistic fuzzymetricwe have 119909 = 119910

Theorem 31 Let (119883M119872119873

T) be a modified intuitionisticfuzzy metric space whereT is a continuous t-representable of


H-type satisfying (22) Let119865 119883times119883 rarr 119883 and 119892 119883 rarr 119883 betwo weakly compatible mappings and there exists 120601 isin Φ suchthat

M119872119873

(119865 (119909 119910) 119865 (119906 V) 120601 (119905))

ge119871lowastT (M

119872119873(119892119909 119892119906 119905) M

119872119873(119892119910 119892V 119905))

(30)

for all 119909 119910 119906 V isin 119883 and 119905 gt 0 Suppose that 119865(119883 times119883) sube 119892(119883)and 119865(119883 times 119883) or 119892(119883) is complete Then there exists a unique119909 isin 119883 such that 119909 = 119892(119909) = 119865(119909 119909)

Proof Let 1199090 1199100isin 119883 be two arbitrary points in 119883 Since

119865(119883 times 119883) sube 119892(119883) we can choose 1199091 1199101isin 119883 such that

1198921199091= 119865 (119909

0 1199100) 119892119910

1= 119865 (119910

0 1199090) (31)

Continuing in this way we can construct two sequences 119909119899

and 119910119899 in119883 such that

119892119909119899+1

= 119865 (119909119899 119910119899) 119892119910

119899+1= 119865 (119910

119899 119909119899) forall119899 ge 0 (32)

The proof is divided into 4 steps

Step 1 Prove that 119892119909119899 and 119892119910

119899 are Cauchy sequences

SinceT is a continuous t-representable of H-type there-fore for any 120582 gt 0 there exists a 120583 gt 0 such that

T119898minus1

(119873119904(120583) 120583) ge

119871lowast (119873119904(120582) 120582) forall119898 isin 119873 (33)

Since lim119905rarrinfin

M119872119873

(119909 119910 119905) = 1119871lowast for all 119909 119910 isin 119883 there

exists 1199050gt 0 such that

M119872119873

(1198921199090 1198921199091 1199050) ge119871lowast (119873119904(120583) 120583)

M119872119873

(1198921199100 1198921199101 1199050) ge119871lowast (119873119904(120583) 120583)

(34)

On the other hand since 120601 isin Φ by condition (120601-3) we havesuminfin

119899=1120601119899(1199050) lt infin Then for any 119905 gt 0 there exists 119899

0isin 119873

such that

119905 gtinfin

sum119896=1198990

120601119896

(1199050) (35)

From condition (30) we have

M119872119873

(1198921199091 1198921199092 120601 (1199050))

= M119872119873

(119865 (1199090 1199100) 119865 (119909

1 1199101) 120601 (119905

0))

ge119871lowastT (M

119872119873(1198921199090 1198921199091 1199050) M119872119873

(1198921199100 1198921199101 1199050))

M119872119873

(1198921199101 1198921199102 120601 (1199050))

= M119872119873

(119865 (1199100 1199090) 119865 (119910

1 1199091) 120601 (119905

0))

ge119871lowastT (M

119872119873(1198921199100 1198921199101 1199050) M119872119873

(1198921199090 1198921199091 1199050))

(36)

Similarly we can also get

M119872119873

(1198921199092 1198921199093 1206012

(1199050))

= M119872119873

(119865 (1199091 1199101) 119865 (119909

2 1199102) 1206012

(1199050))

ge119871lowastT (M

119872119873(1198921199091 1198921199092 120601 (1199050))

M119872119873

(1198921199101 1198921199102 120601 (1199050)))

ge119871lowastT ([M

119872119873(1198921199090 1198921199091 1199050)]2

[M119872119873

(1198921199100 1198921199101 1199050)]2

)

M119872119873

(1198921199102 1198921199103 1206012

(1199050))

= M119872119873

(119865 (1199101 1199091) 119865 (119910

2 1199092) 1206012

(1199050))

ge119871lowastT (M

119872119873(1198921199101 1198921199102 120601 (1199050))

M119872119873

(1198921199091 1198921199092 120601 (1199050)))

ge119871lowastT ([M

119872119873(1198921199100 1198921199101 1199050)]2

[M119872119873

(1198921199090 1198921199091 1199050)]2

)

(37)

Continuing in the same way we can get

M119872119873

(119892119909119899 119892119909119899+1

120601119899

(1199050))

ge119871lowastT ([M

119872119873(1198921199090 1198921199091 1199050)]2119899minus1

[M119872119873

(1198921199100 1198921199101 1199050)]2119899minus1

)

M119872119873

(119892119910119899 119892119910119899+1

120601119899

(1199050))

ge119871lowastT ([M

119872119873(1198921199100 1198921199101 1199050)]2119899minus1

[M119872119873

(1198921199090 1198921199091 1199050)]2119899minus1

)

(38)

Now from (33) (34) and (35) for119898 gt 119899 ge 1198990 we have

M119872119873

(119892119909119899 119892119909119898 119905)

ge119871lowastM119872119873

(119892119909119899 119892119909119898infin

sum119896=1198990

120601119896

(1199050))

ge119871lowastM119872119873

(119892119909119899 119892119909119898119898minus1

sum119896=119899

120601119896

(1199050))

ge119871lowastT119898minus119899minus1

(M119872119873

(119892119909119899 119892119909119899+1

120601119899

(1199050))

M119872119873

(119892119909119899+1

119892119909119899+2

120601119899+1

(1199050))

M119872119873

(119892119909119898minus1

119892119909119898 120601119898minus1

(1199050)))


(T ([M119872119873

(1198921199090 1198921199091 1199050)]2119899minus1

[M119872119873

(1198921199100 1198921199101 1199050)]2119899minus1

)


T ([M119872119873

(1198921199090 1198921199091 1199050)]2119899

[M119872119873

(1198921199100 1198921199101 1199050)]2119899

)

T ([M119872119873

(1198921199090 1198921199091 1199050)]2119898minus2

[M119872119873

(1198921199100 1198921199101 1199050)]2119898minus2

))

ge119871lowastT ([M

119872119873(1198921199090 1198921199091 1199050)]2119898minus1

minus2119899minus1

[M119872119873

(1198921199100 1198921199101 1199050)]2119898minus1

minus2119899minus1

)

ge119871lowastT2119898

minus2119899

minus1

(119873119904(120583) 120583)

ge119871lowast (119873119904(120582) 120582)

(39)

which implies that

M119872119873

(119892119909119899 119892119909119898 119905) gt119871lowast (119873119904(120582) 120582) (40)

for all 119898 119899 isin 119873 with 119898 gt 119899 ge 1198990and 119905 gt 0 So 119892119909

119899 is

a Cauchy sequence Similarly we can get that 119892119910119899 is also a

Cauchy sequence

Step 2 Prove that 119892 and 119865 have a coupled coincidence pointWithout loss of generality we can assume that 119892(119883) is

complete then there exist 119909 119910 isin 119892(119883) and 119886 119887 isin 119883 suchthat

lim119899rarrinfin

119892119909119899= lim119899rarrinfin

119865 (119909119899 119910119899) = 119892119886 = 119909

lim119899rarrinfin

119892119910119899= lim119899rarrinfin

119865 (119910119899 119909119899) = 119892119887 = 119910

(41)

From (30) we get

M119872119873

(119865 (119909119899 119910119899) 119865 (119886 119887) 120601 (119905))

ge119871lowastT (M

119872119873(119892119909119899 119892119886 119905) M

119872119873(119892119910119899 119892119887 119905))

(42)

SinceM119872119873

is continuous taking limit as 119899 rarr infin we have

M119872119873

(119892119886 119865 (119886 119887) 120601 (119905)) = 1119871lowast (43)

which implies that

119865 (119886 119887) = 119892119886 = 119909 (44)

Similarly we can show that

119865 (119887 119886) = 119892119887 = 119910 (45)

Since 119865 and 119892 are weakly compatible we get that

119892119865 (119886 119887) = 119865 (119892119886 119892119887) 119892119865 (119887 119886) = 119865 (119892119887 119892119886) (46)

which implies that

119892119909 = 119865 (119909 119910) 119892119910 = 119865 (119910 119909) (47)

Step 3 Prove that 119892119909 = 119910 and 119892119910 = 119909

SinceT is a continuous t-representable of H-typeThere-fore for any 120582 gt 0 there exists a 120583 gt 0 such that

T119898minus1

(119873119904(120583) 120583) ge



M119872119873



M119872119873

(119892119909 119910 1199050) ge119871lowast (119873119904(120583) 120583)

M119872119873

(119892119910 119909 1199050) ge119871lowast (119873119904(120583) 120583)

(49)



0isin 119873

such that

119905 gtinfin

sum119896=1198990

120601119896

(1199050) (50)

Since

M119872119873

(119892119909 119892119910119899+1

120601 (1199050))

= M119872119873

(119865 (119909 119910) 119865 (119910119899 119909119899) 120601 (119905

0))

ge119871lowastT (M

119872119873(119892119909 119892119910

119899 1199050) lowastM

119872119873(119892119910 119892119909

119899 1199050))

(51)

thus

M119872119873

(119892119909 119892119910119899+1

120601 (1199050))

ge119871lowastT (M

119872119873(119892119909 119892119910

119899 1199050) lowastM

119872119873(119892119910 119892119909

119899 1199050))

(52)

Letting 119899 rarr infin in (52) by using (41) we get

M119872119873

(119892119909 119910 120601 (1199050))

ge119871lowastT (M

119872119873(119892119909 119910 119905

0) M119872119873

(119892119910 119909 1199050))

(53)

Similarly we can get

M119872119873

(119892119910 119909 120601 (1199050))

ge119871lowastT (M

119872119873(119892119909 119910 119905

0) M119872119873

(119892119910 119909 1199050))

(54)

From (53) and (54) we have

T (M119872119873

(119892119909 119910 120601 (1199050)) M

119872119873(119892119910 119909 120601 (119905

0)))

ge119871lowastT ([M

119872119873(119892119909 119910 119905

0)]2

[M119872119873

(119892119910 119909 1199050)]2

) (55)

By this way we can get for all 119899 isin 119873

T (M119872119873

(119892119909 119910 120601119899

(1199050)) M

119872119873(119892119910 119909 120601

119899

(1199050)))

ge119871lowastT ([M

119872119873(119892119909 119910 120601

119899minus1

(1199050))]2

[M119872119873

(119892119910 119909 120601119899minus1

(1199050))]2

)

ge119871lowastT ([M

119872119873(119892119909 119910 119905

0)]2119899

[M119872119873

(119892119910 119909 1199050)]2119899

)

(56)


Thus

T (M119872119873

(119892119909 119910 120601119899

(1199050)) M

119872119873(119892119910 119909 120601

119899

(1199050)))

ge119871lowastT ([M

119872119873(119892119909 119910 119905

0)]2119899

[M119872119873

(119892119910 119909 1199050)]2119899

)

(57)

Then by (48) (49) (50) and (57) we have

T (M119872119873

(119892119909 119910 119905) M119872119873

(119892119910 119909 119905))

ge119871lowastT(M

119872119873(119892119909 119910

infin

sum119896=1198990

120601119896

(1199050))

M119872119873

(119892119910 119909infin

sum119896=1198990

120601119896

(1199050)))

ge119871lowastT (M

119872119873(119892119909 119910 120601

1198990 (1199050)) M

119872119873(119892119910 119909 120601

1198990 (1199050)))

ge119871lowastT ([M

119872119873(119892119909 119910 119905

0)]21198990

[M119872119873

(119892119910 119909 1199050)]21198990

)

ge119871lowastT21198990+1

minus1

(119873119904(120583) 120583)

ge119871lowast (119873119904(120582) 120582)

(58)

So for any 120582 gt 0 we have

T (M119872119873

(119892119909 119910 119905) M119872119873

(119892119910 119909 119905)) ge119871lowast (119873119904(120582) 120582)

forall119905 gt 0

(59)

We can get that 119892119909 = 119910 and 119892119910 = 119909

Step 4 Prove that 119909 = 119910SinceT is a continuous t-representable of H-typeThere-

fore for any 120582 gt 0 there exists a 120583 gt 0 such that

T119898minus1

(119873119904(120583) 120583) ge



M119872119873



M119872119873

(119909 119910 1199050) ge119871lowast (119873119904(120583) 120583) (61)



0isin 119873

such that

119905 gtinfin

sum119896=1198990

120601119896

(1199050) (62)

Since for 1199050gt 0

M119872119873

(119892119909119899+1

119892119910119899+1

120601 (1199050))

= M119872119873

(119865 (119909119899 119910119899) 119865 (119910

119899 119909119899) 120601 (119905

0))

ge119871lowastT (M

119872119873(119892119909119899 119892119910119899 1199050) M119872119873

(119892119910119899 119892119909119899 1199050))

(63)

thus

M119872119873

(119892119909119899+1

119892119910119899+1

120601 (1199050))

ge119871lowastT (M

119872119873(119892119909119899 119892119910119899 1199050)

M119872119873

(119892119910119899 119892119909119899 1199050))

(64)

Letting 119899 rarr infin in the above inequality we get

M119872119873

(119909 119910 120601 (1199050))

ge119871lowastT (M

119872119873(119909 119910 119905

0) M119872119873

(119910 119909 1199050))

(65)

Thus by (60) (61) (62) and (65) we have

M119872119873

(119909 119910 119905)

ge119871lowastM119872119873

(119909 119910infin

sum119896=1198990

120601119896

(1199050))

ge119871lowast M119872119873

(119909 119910 1206011198990 (1199050))

ge119871lowastT ([M

119872119873(119909 119910 119905

0)]21198990minus1

[M119872119873

(119910 119909 1199050)]21198990minus1

)

ge119871lowastT21198990+1

minus3

(119873119904(120583) 120583)

ge119871lowast (119873119904(120582) 120582)

(66)

which implies that 119909 = 119910 Thus we have proved that 119865 and 119892have a unique common fixed point in 119883 This completes theproof of Theorem 31

Taking 119892 = 119868 (the identity mapping) in Theorem 31 weget the following consequence

Corollary 32 Let (119883M119872119873

T) be a complete modifiedintuitionistic fuzzy metric space where T is continuous t-representable of H-type satisfying (22) Let 119865 119883 times 119883 rarr 119883and there exist 120601 isin Φ such that

M119872119873

(119865 (119909 119910) 119865 (119906 V) 120601 (119905))

ge119871lowastT (M

119872119873(119909 119906 119905) M

119872119873(119910 V 119905))

(67)

for all 119909 119910 119906 V isin 119883 and 119905 gt 0 Then there exists 119909 isin 119883 suchthat 119909 = 119865(119909 119909)

Put 120601(119905) = 119896119905 where 0 lt 119896 lt 1 in Theorem 31 we get thefollowing

Corollary 33 Let (119883M119872119873

T) be a modified intuitionisticfuzzy metric space such thatM

119872119873has p-property Let 119865 119883times

119883 rarr 119883 and 119892 119883 rarr 119883 be two functions such that

M119872119873

(119865 (119909 119910) 119865 (119906 V) 119896119905)

ge119871lowastT (M

119872119873(119892119909 119892119906 119905) M

119872119873(119892119910 119892V 119905))

(68)

for all 119909 119910 119906 V isin 119883 and 119905 gt 0 where 0 lt 119896 lt 1 Suppose that119865(119883 times 119883) sube 119892(119883) and 119865(119883 times 119883) or 119892(119883) is complete Thenthere exists a unique 119909 isin 119883 such that 119909 = 119892(119909) = 119865(119909 119909)


Remark 34 ComparingTheorem 31 in the present paperwithTheorem 31 in [13] we can see that Theorem 31 is a genuinegeneralization of Theorem 31 in the sense that

(1) we only use the completeness of 119892(119883) or 119865(119883 times 119883)(2) we drop off the continuity of 119892(3) the concept of compatible mappings has been

replaced by weakly compatible mappings

Next we give an example to demonstrate Theorem 31

Example 35 Let 119883 = 0 1 12 13 1119899 andT(119886 119887) = (min119886

1 1198871max119886

2 1198872) for all 119886 = (119886

1 1198862) and

119887 = (1198871 1198872) isin 119871lowast Define

M119872119873

(119909 119910 119905) = (119905

119905 +1003816100381610038161003816119909 minus 119910

1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119905 +1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)

forall119909 119910 isin 119883 119905 gt 0

(69)

Then (119883M119872119873

T) is a modified intuitionistic fuzzy metricspace Let 120601(119905) = 1199052 Let 119892 119883 rarr 119883 and 119865 119883 times 119883 rarr 119883be defined as

119892 (119909) =

0 119909 = 0

1 119909 =1

2119899 + 1

1

2119899 + 1 119909 =

1

2119899

119865 (119909 119910) =

1

(2119899 + 1)4 (119909 119910) = (

1

21198991

2119899)

0 otherwise

(70)

Let 119909119899= 119910119899= 12119899 We have

119892119909119899=

1

2119899 + 1997888rarr 0

119865 (119909119899 119910119899) =

1

(2119899 + 1)4997888rarr 0 as 119899 997888rarr infin

(71)

but

M119872119873

(119865 (119892119909119899 119892119910119899) 119892119865 (119909

119899 119910119899) 119905)

= M119872119873

(0 1 119905) 999424999426999456 1119871lowast

(72)

so 119892 and 119865 are not compatible From 119865(119909 119910) = 119892(119909) and119865(119910 119909) = 119892(119910) we can get (119909 119910) = (0 0) and we have119892119865(0 0) = 119865(1198920 1198920) which implies that 119865 and 119892 are weaklycompatible The following result is easy to be verified

119905

119883 + 119905ge min 119905

119884 + 119905

119905

119885 + 119905

119883

119883 + 119905le max 119884

119884 + 119905

119885

119885 + 119905

lArrrArr 119883 le max 119884 119885 forall119883 119884 119885 ge 0 119905 gt 0

(73)

By the definition ofM119872119873

120601 and the result above we can getinequality (30)

M119872119873

(119865 (119909 119910) 119865 (119906 V) 120601 (119905))

ge119871lowastT (M

119872119873(119892119909 119892119906 119905) M

119872119873(119892119910 119892V 119905))

(74)

which is equivalent to the following

21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816 le max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V

1003816100381610038161003816 (75)

Now we verify inequality (75) Let 119860 = 12119899 119899 isin 119873 119861 =119883 minus 119860 By the symmetry and without loss of generality(119909 119910) (119906 V) have six possibilities

Case 1 (119909 119910) isin 119861 times 119861 (119906 V) isin 119861 times 119861 It is obvious that (75)holds


Case 3 (119909 119910) isin 119861 times 119861 (119906 V) isin 119860 times 119860 If 119906 = V (75) holds If119906 = V let 119906 = V = 12119899 and then

21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816 =2

(2119899 + 1)4

max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V

1003816100381610038161003816 =2119899

2119899 + 1

(76)

which implies that (75) holds


Case 5 (119909 119910) isin 119861 times 119860 (119906 V) isin 119860 times 119860 If 119906 = V (75) holds If119906 = V let 119909 isin 119861 119910 = 12119895 119906 = V = 12119899 and then

21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816 =2

(2119899 + 1)4

max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V

1003816100381610038161003816

= max 1

2119899 + 110038161003816100381610038161003816100381610038161003816

1

2119895 + 1minus

1

2119899 + 1

10038161003816100381610038161003816100381610038161003816

(77)

or

max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V

1003816100381610038161003816

= max 2119899

2119899 + 110038161003816100381610038161003816100381610038161003816

1

2119895 + 1minus

1

2119899 + 1

10038161003816100381610038161003816100381610038161003816

(78)

So (75) holds

Case 6 (119909 119910) isin 119860 times 119860 and (119906 V) isin 119860 times 119860 If 119909 = 119910 and 119906 = V(75) holds If 119909 = 119910 119906 = V let 119909 = 12119894 119910 = 12119895 119894 = 119895 and119906 = V = 12119899 Then

21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816 =2

(2119899 + 1)4

max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V

1003816100381610038161003816

= max1003816100381610038161003816100381610038161003816

1

2119894 + 1minus

1

2119899 + 1

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1

2119895 + 1minus

1

2119899 + 1

10038161003816100381610038161003816100381610038161003816

(79)


So (75) holds If 119909 = 119910 and 119906 = V let 119909 = 119910 = 12119894 119906 = V =12119899 Then

21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816 = 2

100381610038161003816100381610038161003816100381610038161003816

1

(2119894 + 1)4minus

1

(2119899 + 1)4

100381610038161003816100381610038161003816100381610038161003816

max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V

1003816100381610038161003816

=1

2119894 + 1minus

1

2119899 + 1

(80)

So (75) holds Then all the conditions in Theorem 31 aresatisfied and 0 is the unique common fixed point of 119892 and119865

4 Application to Integral Equations

As an application of the coupled fixed point theorems estab-lished in Section 3 of our paper we study the existence anduniqueness of the solution to a Fredholm nonlinear integralequation We will consider the following integral equation

119909 (119901) = int119887

119886

(1198701(119901 119902) + 119870

2(119901 119902))

times [119891 (119902 119909 (119902)) + 119892 (119902 119909 (119902))] 119889119902 + ℎ (119901)

(81)

for all 119901 isin 119868 = [119886 119887]Let Θ denote the set of all functions 120579 [0 +infin) rarr

[0 +infin) satisfying the following

(i120579) 120579 is nondecreasing

(ii120579) 120579(119901) le 119901

We assume that the functions 1198701 1198702 119891 119892 fulfill the

following conditions

Assumption 36 (i) Consider

1198701(119901 119902) ge 0 119870

2(119901 119902) le 0 forall119901 119902 isin 119868 (82)

(ii) There exists positive numbers 120582 120583 and 120579 isin Θ suchthat for all 119909 119910 isin 119877 with 119909 ge 119910 the following conditionshold

0 le 119891 (119902 119909) minus 119891 (119902 119910) le 120582120579 (119909 minus 119910)

minus120583120579 (119909 minus 119910) le 119892 (119902 119909) minus 119892 (119902 119910) le 0(83)

(iii) Consider

max 120582 120583 sup119901isin119868

int119887

119886

[1198701(119901 119902) minus 119870

2(119901 119902)] 119889119902 le

1

8 (84)

Theorem 37 Consider the integral equation (81) with1198701 1198702isin 119862(119868 times 119868R) 119891 119892 isin 119862(119868 times RR) and ℎ isin 119862(119868 119877)

Suppose that Assumption 36 is satisfied Then the integralequation (81) has a unique solution in 119862(119868 119877)

Proof Consider 119883 = 119862(119868 119877) It is easy to check that(119883M

119872119873T) is a complete modified intuitionistic fuzzy

metric space with respect to the modified intuitionistic fuzzymetric

M119872119873

(119909 119910 119905) = (119905

119905 +1003816100381610038161003816119909 minus 119910

1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119905 +1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)

forall119909 119910 isin 119883 119905 gt 0

(85)

for all 119909 119910 isin 119883 and 119905 gt 0 with T(119886 119887) =(min119886

1 1198871max119886

2 1198872) for all 119886 = (119886

1 1198862) and 119887 = (119887

1 1198872) isin

119871lowast Define now the mapping 119865 119883 times 119883 rarr 119883 by

119865 (119909 119910) (119901) = int119887

119886

1198701(119901 119902) [119891 (119902 119909 (119902)) + 119892 (119902 119910 (119902))] 119889119902

+ int119887

119886

1198702(119901 119902) [119891 (119902 119910 (119902))

+119892 (119902 119909 (119902))] 119889119902 + ℎ (119901)

(86)

for all119901 isin 119868 and120601(119905) = 1199052 for all 119905 gt 0 Now for all119909 119910 119906 V isin119883 using (83) we have

119865 (119909 119910) (119901) minus 119865 (119906 V) (119901)

= int119887

119886

1198701(119901 119902) [119891 (119902 119909 (119902)) + 119892 (119902 119910 (119902))] 119889119902

+ int119887

119886

1198702(119901 119902) [119891 (119902 119910 (119902)) + 119892 (119902 119909 (119902))] 119889119902

minus int119887

119886

1198701(119901 119902) [119891 (119902 119906 (119902)) + 119892 (119902 V (119902))] 119889119902

minus int119887

119886

1198702(119901 119902) [119891 (119902 V (119902)) + 119892 (119902 119906 (119902))] 119889119902

= int119887

119886

1198701(119901 119902) [119891 (119902 119909 (119902)) minus 119891 (119902 119906 (119902))

+119892 (119902 119910 (119902)) minus 119892 (119902 V (119902))] 119889119902

+ int119887

119886

1198702(119901 119902) [119891 (119902 119910 (119902)) minus 119891 (119902 V (119902))

+119892 (119902 119909 (119902)) minus 119892 (119902 119906 (119902))] 119889119902

= int119887

119886

1198701(119901 119902) [(119891 (119902 119909 (119902)) minus 119891 (119902 119906 (119902)))

minus (119892 (119902 V (119902)) minus 119892 (119902 119910 (119902)))] 119889119902

minus int119887

119886

1198702(119901 119902) [(119891 (119902 V (119902)) minus 119891 (119902 119910 (119902)))

minus (119892 (119902 119909 (119902)) minus 119892 (119902 119906 (119902)))] 119889119902

le int119887

119886

1198701(119901 119902) [120582120579 (119909 (119902) minus 119906 (119902))

+120583120579 (V (119902) minus 119910 (119902))] 119889119902


minus int119887

119886

1198702(119901 119902) [120582120579 (V (119902) minus 119910 (119902))

+120583120579 (119909 (119902) minus 119906 (119902))] 119889119902

(87)

Thus

119865 (119909 119910) (119901) minus 119865 (119906 V) (119901)

le int119887

119886

1198701(119901 119902) [120582120579 (119909 (119902) minus 119906 (119902))

+120583120579 (V (119902) minus 119910 (119902))] 119889119902

minus int119887

119886

1198702(119901 119902) [120582120579 (V (119902) minus 119910 (119902))

+120583120579 (119909 (119902) minus 119906 (119902))] 119889119902

(88)

Since the function 120579 is nondecreasing so we have

120579 (119909 (119902) minus 119906 (119902)) le 120579 (1003816100381610038161003816119909 (119902) minus 119906 (119902)

1003816100381610038161003816)

120579 (V (119902) minus 119910 (119902)) le 120579 (1003816100381610038161003816V (119902) minus 119910 (119902)

1003816100381610038161003816) (89)

Hence by (88) in view of the fact that 1198702(119901 119902) le 0 we get

1003816100381610038161003816119865 (119909 119910) (119901) minus 119865 (119906 V) (119901)1003816100381610038161003816

le int119887

119886

1198701(119901 119902) [120582120579 (

1003816100381610038161003816119909 (119902) minus 119906 (119902)1003816100381610038161003816)

+120583120579 (1003816100381610038161003816V (119902) minus 119910 (119902)

1003816100381610038161003816)] 119889119902

minus int119887

119886

1198702(119901 119902) [120582120579 (

1003816100381610038161003816V (119902) minus 119910 (119902)1003816100381610038161003816)

+120583120579 (1003816100381610038161003816119909 (119902) minus 119906 (119902)

1003816100381610038161003816)] 119889119902

le int119887

119886

1198701(119901 119902) [max 120582 120583 120579 (1003816100381610038161003816119909 (119902) minus 119906 (119902)

1003816100381610038161003816)

+max 120582 120583 120579 (1003816100381610038161003816V (119902) minus 119910 (119902)1003816100381610038161003816)] 119889119902

minus int119887

119886

1198702(119901 119902) [max 120582 120583 120579 (1003816100381610038161003816V (119902) minus 119910 (119902)

1003816100381610038161003816)

+max 120582 120583 120579 (1003816100381610038161003816119909 (119902) minus 119906 (119902)1003816100381610038161003816)] 119889119902

(90)

as all the quantities on the right hand side of (88) arenonnegative Now by using (84) we get

1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)1003816100381610038161003816

le max 120582 120583 int119887

119886

[1198701(119901 119902) minus 119870

2(119901 119902)] 119889119902

sdot [120579 (1003816100381610038161003816119909 (119902)minus119906 (119902)

1003816100381610038161003816)+120579 (1003816100381610038161003816V (119902) minus 119910 (119902)

1003816100381610038161003816)]

le max 120582 120583 sup119901isin119868

int119887

119886

[1198701(119901 119902) minus 119870

2(119901 119902)] 119889119902

sdot [120579 (1003816100381610038161003816119909 (119902)minus119906 (119902)

1003816100381610038161003816)+120579 (1003816100381610038161003816V (119902) minus 119910 (119902)

1003816100381610038161003816)]

le120579 (|119909 minus 119906|) + 120579 (

1003816100381610038161003816V minus 1199101003816100381610038161003816)

8

(91)

Thus

21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816 le120579 (|119909 minus 119906|) + 120579 (

1003816100381610038161003816V minus 1199101003816100381610038161003816)

4 (92)

Now since 120579 is nondecreasing we have

120579 (|119909 minus 119906|) le 120579 (|119909 minus 119906| +1003816100381610038161003816119910 minus V1003816100381610038161003816)

120579 (1003816100381610038161003816119910 minus V1003816100381610038161003816) le 120579 (|119909 minus 119906| +

1003816100381610038161003816119910 minus V1003816100381610038161003816) (93)

which by (ii120579) implies

120579 (|119909 minus 119906|) + 120579 (1003816100381610038161003816119910 minus V1003816100381610038161003816)

2le 120579 (|119909 minus 119906| +

1003816100381610038161003816119910 minus V1003816100381610038161003816)

le |119909 minus 119906| +1003816100381610038161003816119910 minus V1003816100381610038161003816

le 2max |119909 minus 119906| 1003816100381610038161003816119910 minus V1003816100381610038161003816

(94)

and so

120579 (|119909 minus 119906|) + 120579 (1003816100381610038161003816119910 minus V1003816100381610038161003816)

4le max |119909 minus 119906| 1003816100381610038161003816119910 minus V1003816100381610038161003816 (95)

Thus by (92) and (95) we get

21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816 le max |119909 minus 119906| 1003816100381610038161003816119910 minus V1003816100381610038161003816 (96)

Now by (96) and (73) it follows that

M119872119873

(119865 (119909 119910) 119865 (119906 V) 120601 (119905))

= M119872119873

(119865 (119909 119910) 119865 (119906 V) 119905

2)

= (1199052

1199052 +1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816

1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)1003816100381610038161003816

1199052 +1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816)

= (119905

119905 + 21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816

21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816119905 + 2

1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)1003816100381610038161003816)

ge119871lowast (

119905

119905 +max |119909 minus 119906| 1003816100381610038161003816119910 minus V1003816100381610038161003816

max |119909 minus 119906| 1003816100381610038161003816119910 minus V1003816100381610038161003816119905 +max |119909 minus 119906| 1003816100381610038161003816119910 minus V1003816100381610038161003816

)


ge119871lowast (min 119905

119905 + |119909 minus 119906|

119905

119905 +1003816100381610038161003816119910 minus V1003816100381610038161003816

max |119909 minus 119906|

119905 + |119909 minus 119906|

1003816100381610038161003816119910 minus V1003816100381610038161003816119905 +

1003816100381610038161003816119910 minus V1003816100381610038161003816)

ge119871lowastT (M

119872119873(119909 119906 119905) M

119872119873(119910 V 119905))

(97)

which is the condition in (67) shows that all hypotheses ofCorollary 32 are satisfied This proves that 119865 has a uniquefixed point 119909 isin 119883 that is 119909 = 119865(119909 119909) and therefore119909 isin 119862(119868 119877) is the unique solution of the integral equation(81)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] I Kramosil and JMichalek ldquoFuzzymetric and statistical metricspacesrdquo Kybernetika vol 11 no 5 pp 336ndash344 1975

[2] A George and P Veeramani ldquoOn some results in fuzzy metricspacesrdquo Fuzzy Sets and Systems vol 64 no 3 pp 395ndash399 1994

[3] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986

[4] C Alaca D Turkoglu and C Yildiz ldquoFixed points in intuition-istic fuzzy metric spacesrdquo Chaos Solitons amp Fractals vol 29 no5 pp 1073ndash1078 2006

[5] J H Park ldquoIntuitionistic fuzzy metric spacesrdquo Chaos Solitonsamp Fractals vol 22 no 5 pp 1039ndash1046 2004

[6] V Gregori S Romaguera and P Veeramani ldquoA note onintuitionistic fuzzy metric spacesrdquo Chaos Solitons amp Fractalsvol 28 no 4 pp 902ndash905 2006

[7] R Saadati S Sedghi and N Shobe ldquoModified intuitionisticfuzzy metric spaces and some fixed point theoremsrdquo ChaosSolitons amp Fractals vol 38 no 1 pp 36ndash47 2008

[8] T G Bhaskar and V Lakshmikantham ldquoFixed point theoremsin partially ordered metric spaces and applicationsrdquo NonlinearAnalysis Theory Methods and Applications vol 65 no 7 pp1379ndash1393 2006

[9] V Lakshmikantham and L Ciric ldquoCoupled fixed point the-orems for nonlinear contractions in partially ordered metricspacesrdquo Nonlinear Analysis Theory Methods and Applicationsvol 70 no 12 pp 4341ndash4349 2009

[10] S Sedghi I Altun andN Shobe ldquoCoupled fixedpoint theoremsfor contractions in fuzzy metric spacesrdquo Nonlinear AnalysisTheory Methods and Applications vol 72 no 3-4 pp 1298ndash1304 2010

[11] X Q Hu ldquoCommon coupled fixed point theorems for contrac-tive mappings in fuzzy metric spacesrdquo Fixed Point Theory andApplications vol 2011 Article ID 363716 14 pages 2011

[12] X Q Hu M X Zheng B Damjanovic and X F ShaoldquoCommon coupled fixed point theorems for weakly compatiblemappings in fuzzy metric spacesrdquo Fixed Point Theory andApplications vol 2013 article 220 2013

[13] B Deshpande S Sharma and A Handa ldquoCommon coupledfixed point theorems for nonlinear contractive condition onintuitionistic fuzzy metric space with application to integral

equationsrdquo Journal of the Korean Society of MathematicalEducation B The Pure and Applied Mathematics vol 20 no 3pp 159ndash180 2013

[14] G Deschrijver and E E Kerre ldquoOn the relationship betweensome extensions of fuzzy set theoryrdquo Fuzzy Sets and Systemsvol 133 no 2 pp 227ndash235 2003

[15] G Deschrijver C Cornelis and E E Kerre ldquoOn the repre-sentation of intuitionistic fuzzy t-norms and t-conormsrdquo IEEETransactions on Fuzzy Systems vol 12 no 1 pp 45ndash61 2004

[16] J X Fang ldquoOn fixed point theorems in fuzzy metric spacesrdquoFuzzy Sets and Systems vol 46 no 1 pp 107ndash113 1992

[17] R Saadati and J H Park ldquoOn the modified intuitionistic fuzzytopological spacesrdquo Chaos Solitons amp Fractals vol 27 no 2 pp331ndash334 2006

[18] J X Fang ldquoCommon fixed point theorems of compatible andweakly compatible maps inMenger spacesrdquoNonlinear AnalysisTheoryMethods andApplications vol 71 no 5-6 pp 1833ndash18432009

[19] M Abbas M A Khan and S Radenovic ldquoCommon coupledfixed point theorems in cone metric spaces for w-compatiblemappingsrdquo Applied Mathematics and Computation vol 217 no1 pp 195ndash202 2010

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they study the existence and uniqueness of the solution to anonlinear Fredholm integral equation

In this paper we prove a common coupled fixed pointtheorem for weakly compatible mappings on modifiedintuitionistic fuzzy metric spaces As an application of ourresult we study the existence and uniqueness of the solutionto a nonlinear Fredholm integral equation which arisenaturally in the theory of signal processing linear forwardmodeling and inverse problems We also give an example tovalidate our result We extend and generalize the results ofHu [11] Hu et al [12] and Sedghi et al [10] in the settings ofmodified intuitionistic fuzzy metric spaces The result is thegenuine generalization of the result of Deshpande et al [13]

2 Preliminaries

Lemma 1 (Deschrijver and Kerre [14]) Consider the set 119871lowastand operation le

119871lowastdefined by

119871lowast

= (1199091 1199092) (1199091 1199092) isin [0 1]

2

1199091+ 1199092le 1 (1)

(1199091 1199092)le119871lowast(1199101 1199102) hArr 119909

1le 1199101 and 119909

2ge 1199102for every

(1199091 1199092) (1199101 1199102) isin 119871lowast Then (119871lowast le

119871lowast) is a complete lattice

Definition 2 (Atanassov [3]) An intuitionistic fuzzy setA120577120578

in a universe 119880 is an object A120577120578

= 120577A(119906) 120578A(119906) wherefor all 119906 isin 119880 120577A(119906) isin [0 1] and 120578A(119906) isin [0 1] arecalled the membership degree and nonmembership degreerespectively of 119906 in A

120577120578and further they satisfy 120577A(119906) +

120578A(119906) le 1 For every 119911119894= (119909119894 119910119894) isin 119871lowast if 119888

119894isin [0 1] such

that sum119899119895=1

119888119895= 1 then it is easy to see that

1198881(1199091 1199101) + sdot sdot sdot + 119888

119899(119909119899 119910119899)

=119899

sum119895=1

119888119895(119909119895 119910119895) = (

119899

sum119895=1

119888119895119909119895119899

sum119895=1

119888119895119910119895) isin 119871

lowast

(2)

We denote its units by 0119871lowast = (0 1) and 1

119871lowast = (1 0)

Classically a triangular norm lowast = 119879 on [0 1] is defined as anincreasing commutative associative mapping 119879 [0 1]2 rarr[0 1] satisfying 119879(1 119909) = 1 lowast 119909 = 119909 for all 119909 isin [0 1]A triangular conorm 119878 = is defined as an increasingcommutative associative mapping 119878 [0 1]2 rarr [0 1]satisfying 119878(0 119909) = 0 119909 = 119909 for all 119909 isin [0 1] Usingthe lattice (119871lowast le

119871lowast) these definitions can be straightforwardly

extended

Definition 3 (Deschrijver et al [15]) A triangular norm (t-norm) on 119871lowast is a mapping T (119871lowast)2 rarr 119871lowast satisfying thefollowing conditions

(forall119909 isin 119871lowast)(T(119909 1119871lowast) = 119909) (boundary condition)

(forall(119909 119910) isin (119871lowast)2

) (T(119909 119910) = T(119910 119909)) (commutativ-ity)(forall(119909 119910 119911) isin (119871lowast)

3

) (T(119909T(119910 119911)) = T(T(119909 119910) 119911))(associativity)(forall(119909 1199091015840 119910 1199101015840) isin (119871lowast)

4

) (119909le119871lowast1199091015840 and 119910le

119871lowast1199101015840 rArr

T(119909 119910)le119871lowastT(1199091015840 1199101015840)) (monotonicity)

Definition 4 (Deschrijver and Kerre [14] and Deschrijver etal [15]) A continuous t-normT on 119871lowast is called continuoust-representable if and only if there exist a continuous t-normlowast and a continuous t-conorm on [0 1] such that for all119909 = (119909

1 1199092) 119910 = (119910

1 1199102) isin 119871lowast

T (119909 119910) = (1199091lowast 1199101 1199092 1199102) (3)

Now define a sequenceT119899 recursively byT1 = T and

T119899

(119909(1)

119909(119899+1)

) = T (T119899minus1

(119909(1)

119909(119899)

) 119909(119899+1)

)

(4)

for 119899 ge 2 and 119909(119894) isin 119871lowast

Definition 5 (Deschrijver and Kerre [14] andDeschrijver et al[15]) A negator on 119871lowast is any decreasing mappingN 119871lowast rarr119871lowast satisfyingN(0

119871lowast) = 1

119871lowast andN(1

119871lowast) = 0

119871lowast IfN(N(119909)) =

119909 for all 119909 isin 119871lowast then N is called an involutive negator Anegator on [0 1] is a decreasing mapping119873 [0 1] rarr [0 1]satisfying 119873(0) = 1 and 119873(1) = 0 119873

119904denotes the standard

negator on [0 1] defined as for all 119909 isin [0 1]119873119904(119909) = 1 minus 119909

Definition 6 (Saadati et al [7]) Let119872119873 be fuzzy sets from1198832 times (0 + infin) to [0 1] such that 119872(119909 119910 119905) + 119873(119909 119910 119905) le 1for all 119909 119910 isin 119883 and 119905 gt 0 The 3-tuple (119883 M

119872119873T) is said

to be a modified intuitionistic fuzzy metric space if 119883 is anarbitrary (nonempty) setT is a continuous t-representableandM

119872119873is a mapping 1198832 times (0 + infin) rarr 119871lowast satisfying the

following conditions for every 119909 119910 119911 isin 119883 and 119905 119904 gt 0

(a) M119872119873

(119909 119910 119905)gt119871lowast0119871lowast

(b) M119872119873

(119909 119910 119905) = 1119871lowast if and only if 119909 = 119910

(c) M119872119873

(119909 119910 119905) = M119872119873

(119910 119909 119905)(d) M

119872119873(119909 119910 119905+119904)ge

119871lowastT(M

119872119873(119909 119911 119905)M

119872119873(119911 119910 119904))

(e) M119872119873

(119909 119910 sdot) (0 infin) rarr 119871lowast is continuous

In this caseM119872119873

is called amodified intuitionistic fuzzymetric Here

M119872119873

(119909 119910 119905) = (119872(119909 119910 119905)119873(119909 119910 119905))

Example 7 (Saadati et al [7]) Let (119883 119889) be a metric spaceDenoteT(119886 119887) = (119886

11198871min119886

2+1198872 1) for all 119886 = (119886

1 1198862) and


defined as follows

M119872119873

(119909 119910 119905) = (119872 (119909 119910 119905) 119873 (119909 119910 119905))

= (ℎ119905119899

ℎ119905119899 + 119898119889 (119909 119910)

119898119889 (119909 119910)

ℎ119905119899 + 119898119889 (119909 119910))

(5)

for all 119905 ℎ 119898 119899 isin 119877+ Then (119883M119872119873


Example 8 (Saadati et al [7]) Let 119883 = 119873 DenoteT(119886 119887) =(max0 119886

1+ 1198871minus 1 119886

2+ 1198872minus 11988621198872) for all 119886 = (119886

1 1198862) and



defined as follows

M119872119873

(119909 119910 119905) = (119872 (119909 119910 119905) 119873 (119909 119910 119905))

=

(119909

119910119910 minus 119909

119910) if 119909 le 119910

(119910

119909119909 minus 119910

119909) if 119910 le 119909

(6)







119871lowast







M119872119873

(119909119899 119909119898 119905) gt119871lowast (119873119904(120576) 120576) (7)





T) and

denoted by 119909119899

M119872119873

997888997888997888997888rarr 119909 if M119872119873

(119909119899 119909 119905) rarr 1





119872119873(119909 119910 119905)




119861 (119909 119903 119905) = 119910 isin 119883 M119872119873

(119909 119910 119905) gt119871lowast (119873119904(119903) 119903) (8)






lim119899rarrinfin

M119872119873

(119909119899 119910119899 119905119899) = M

119872119873(119909 119910 119905) (9)



lim119899rarrinfin

M119872119873

(119909119899 119910 119905) = M

119872119873(119909 119910 119905)

lim119899rarrinfin

M119872119873

(119909 119910119899 119905) = M

119872119873(119909 119910 119905)

lim119899rarrinfin

M119872119873

(119909 119910 119905119899) = M

119872119873(119909 119910 119905)

(10)




119865 (119909 119910) = 119909 119865 (119910 119909) = 119910 (11)


119865 (119909 119910) = 119892 (119909) 119865 (119910 119909) = 119892 (119910) (12)


119909 = 119865 (119909 119910) = 119892 (119909) 119910 = 119865 (119910 119909) = 119892 (119910) (13)


119909 = 119865 (119909 119909) = 119892 (119909) (14)





lim119899rarrinfin

M119872119873

(119892119865 (119909119899 119910119899) 119865 (119892119909

119899 119892119910119899) 119905) = 1

119871lowast

lim119899rarrinfin

M119872119873

(119892119865 (119910119899 119909119899) 119865 (119892119910

119899 119892119909119899) 119905) = 1

119871lowast

(16)




119865 (119909119899 119910119899) = lim119899rarrinfin

119892 (119909119899) = 119909

lim119899rarrinfin

119865 (119910119899 119909119899) = lim119899rarrinfin

119892 (119910119899) = 119910 forall119909 119910 isin 119883

(17)



3 Main Results






T1

(119905) = T (119905 119905) T119898+1

(119905) = T (119905T119898

(119905))

119898 = 1 2 119905 isin 119871lowast

(18)


T119898


119871lowast (119873119904(120583) 120583)

(19)



119872119873(119909 119910 119905)gt

119871lowast(119873119904(119903) 119903) for


0lt 119905

such that M119872119873

(119909 119910 1199050) gt (119873

119904(119903) 119903)


[M119872119873

(119909 119910 119905)]119899

= T119899minus1

(M119872119873

(119909 119910 119905)) (20)

for all 119899 isin 119873

Definition 26 Let (119883M119872119873




lim119901rarrinfin

[M119872119873

(119909 119910 119896119901

119905)]119901

= 1119871lowast (21)


Lemma 27 Let (119883M119872119873



lim119905rarrinfin

M119872119873





lim119905rarrinfin

M119872119873

(1199090 1199100 119905) = (120582119873

119904(120582)) lt 1

119871lowast (23)


lim119901rarrinfin

[M119872119873

(1199090 1199100 119896119901

119905)]119901

= 0119871lowast (24)




120595 (119905) =

120572radic119905 0 le 119905 le 4

1 minus1

ln 119905 119905 gt 4

(25)


119905rarrinfin120595(119905) = 1 Let

M119872119873

(119909 119910 119905) = ([120595 (119905)]119889(119909119910)

1 minus [120595 (119905)]119889(119909119910)

) (26)


for all 119886 = (1198861 1198862) and 119887 = (119887


119872119873T)


lim119905rarrinfin

M119872119873

(119909 119910 119905)


([120595 (119905)]119889(119909119910)

1 minus [120595 (119905)]119889(119909119910)

)


(27)


lim119901rarrinfin

[M119872119873

(119909 119910 119896119901

119905)]119901


[120595 (119896119901

119905)]119889(119909119910)sdot119901

lim119901rarrinfin

[1 minus [120595 (119896119901

119905)]119889(119909119910)sdot119901

])


[1 minus1

ln (119896119901119905)]119889(119909119910)sdot119901

lim119901rarrinfin

[1 minus [1 minus1

ln(119896119901119905)]119901sdot119889(119909119910)

])

= (119890minus119889(119909119910) ln 119896

1 minus 119890minus119889(119909119910) ln 119896

)

= 1119871lowast

(28)




120601119899(120601(119905)) 119899 isin 119873


Lemma 30 Let (119883M119872119873


M119872119873

(119909 119910 120601 (119905)) ge119871lowastM119872119873

(119909 119910 119905) forall119905 gt 0 (29)

then 119909 = 119910


we have M119872119873

(119909 119910 119905)ge119871lowastM119872119873


Theorem 31 Let (119883M119872119873




M119872119873

(119865 (119909 119910) 119865 (119906 V) 120601 (119905))

ge119871lowastT (M

119872119873(119892119909 119892119906 119905) M

119872119873(119892119910 119892V 119905))

(30)




1198921199091= 119865 (119909

0 1199100) 119892119910

1= 119865 (119910

0 1199090) (31)


and 119910119899 in119883 such that

119892119909119899+1

= 119865 (119909119899 119910119899) 119892119910

119899+1= 119865 (119910

119899 119909119899) forall119899 ge 0 (32)





T119898minus1

(119873119904(120583) 120583) ge



M119872119873



M119872119873

(1198921199090 1198921199091 1199050) ge119871lowast (119873119904(120583) 120583)

M119872119873

(1198921199100 1198921199101 1199050) ge119871lowast (119873119904(120583) 120583)

(34)



0isin 119873

such that

119905 gtinfin

sum119896=1198990

120601119896

(1199050) (35)


M119872119873

(1198921199091 1198921199092 120601 (1199050))

= M119872119873

(119865 (1199090 1199100) 119865 (119909

1 1199101) 120601 (119905

0))

ge119871lowastT (M

119872119873(1198921199090 1198921199091 1199050) M119872119873

(1198921199100 1198921199101 1199050))

M119872119873

(1198921199101 1198921199102 120601 (1199050))

= M119872119873

(119865 (1199100 1199090) 119865 (119910

1 1199091) 120601 (119905

0))

ge119871lowastT (M

119872119873(1198921199100 1198921199101 1199050) M119872119873

(1198921199090 1198921199091 1199050))

(36)


M119872119873

(1198921199092 1198921199093 1206012

(1199050))

= M119872119873

(119865 (1199091 1199101) 119865 (119909

2 1199102) 1206012

(1199050))

ge119871lowastT (M

119872119873(1198921199091 1198921199092 120601 (1199050))

M119872119873

(1198921199101 1198921199102 120601 (1199050)))

ge119871lowastT ([M

119872119873(1198921199090 1198921199091 1199050)]2

[M119872119873

(1198921199100 1198921199101 1199050)]2

)

M119872119873

(1198921199102 1198921199103 1206012

(1199050))

= M119872119873

(119865 (1199101 1199091) 119865 (119910

2 1199092) 1206012

(1199050))

ge119871lowastT (M

119872119873(1198921199101 1198921199102 120601 (1199050))

M119872119873

(1198921199091 1198921199092 120601 (1199050)))

ge119871lowastT ([M

119872119873(1198921199100 1198921199101 1199050)]2

[M119872119873

(1198921199090 1198921199091 1199050)]2

)

(37)


M119872119873

(119892119909119899 119892119909119899+1

120601119899

(1199050))

ge119871lowastT ([M

119872119873(1198921199090 1198921199091 1199050)]2119899minus1

[M119872119873

(1198921199100 1198921199101 1199050)]2119899minus1

)

M119872119873

(119892119910119899 119892119910119899+1

120601119899

(1199050))

ge119871lowastT ([M

119872119873(1198921199100 1198921199101 1199050)]2119899minus1

[M119872119873

(1198921199090 1198921199091 1199050)]2119899minus1

)

(38)


M119872119873

(119892119909119899 119892119909119898 119905)

ge119871lowastM119872119873

(119892119909119899 119892119909119898infin

sum119896=1198990

120601119896

(1199050))

ge119871lowastM119872119873

(119892119909119899 119892119909119898119898minus1

sum119896=119899

120601119896

(1199050))


(M119872119873

(119892119909119899 119892119909119899+1

120601119899

(1199050))

M119872119873

(119892119909119899+1

119892119909119899+2

120601119899+1

(1199050))

M119872119873

(119892119909119898minus1

119892119909119898 120601119898minus1

(1199050)))


(T ([M119872119873

(1198921199090 1198921199091 1199050)]2119899minus1

[M119872119873

(1198921199100 1198921199101 1199050)]2119899minus1

)


T ([M119872119873

(1198921199090 1198921199091 1199050)]2119899

[M119872119873

(1198921199100 1198921199101 1199050)]2119899

)

T ([M119872119873

(1198921199090 1198921199091 1199050)]2119898minus2

[M119872119873

(1198921199100 1198921199101 1199050)]2119898minus2

))

ge119871lowastT ([M

119872119873(1198921199090 1198921199091 1199050)]2119898minus1

minus2119899minus1

[M119872119873

(1198921199100 1198921199101 1199050)]2119898minus1

minus2119899minus1

)


minus2119899

minus1

(119873119904(120583) 120583)

ge119871lowast (119873119904(120582) 120582)

(39)

which implies that

M119872119873

(119892119909119899 119892119909119898 119905) gt119871lowast (119873119904(120582) 120582) (40)


119899 is


Cauchy sequence



lim119899rarrinfin

119892119909119899= lim119899rarrinfin

119865 (119909119899 119910119899) = 119892119886 = 119909

lim119899rarrinfin

119892119910119899= lim119899rarrinfin

119865 (119910119899 119909119899) = 119892119887 = 119910

(41)

From (30) we get

M119872119873

(119865 (119909119899 119910119899) 119865 (119886 119887) 120601 (119905))

ge119871lowastT (M

119872119873(119892119909119899 119892119886 119905) M

119872119873(119892119910119899 119892119887 119905))

(42)

SinceM119872119873


M119872119873

(119892119886 119865 (119886 119887) 120601 (119905)) = 1119871lowast (43)

which implies that

119865 (119886 119887) = 119892119886 = 119909 (44)


119865 (119887 119886) = 119892119887 = 119910 (45)


119892119865 (119886 119887) = 119865 (119892119886 119892119887) 119892119865 (119887 119886) = 119865 (119892119887 119892119886) (46)

which implies that

119892119909 = 119865 (119909 119910) 119892119910 = 119865 (119910 119909) (47)



T119898minus1

(119873119904(120583) 120583) ge



M119872119873



M119872119873

(119892119909 119910 1199050) ge119871lowast (119873119904(120583) 120583)

M119872119873

(119892119910 119909 1199050) ge119871lowast (119873119904(120583) 120583)

(49)



0isin 119873

such that

119905 gtinfin

sum119896=1198990

120601119896

(1199050) (50)

Since

M119872119873

(119892119909 119892119910119899+1

120601 (1199050))

= M119872119873

(119865 (119909 119910) 119865 (119910119899 119909119899) 120601 (119905

0))

ge119871lowastT (M

119872119873(119892119909 119892119910

119899 1199050) lowastM

119872119873(119892119910 119892119909

119899 1199050))

(51)

thus

M119872119873

(119892119909 119892119910119899+1

120601 (1199050))

ge119871lowastT (M

119872119873(119892119909 119892119910

119899 1199050) lowastM

119872119873(119892119910 119892119909

119899 1199050))

(52)


M119872119873

(119892119909 119910 120601 (1199050))

ge119871lowastT (M

119872119873(119892119909 119910 119905

0) M119872119873

(119892119910 119909 1199050))

(53)


M119872119873

(119892119910 119909 120601 (1199050))

ge119871lowastT (M

119872119873(119892119909 119910 119905

0) M119872119873

(119892119910 119909 1199050))

(54)


T (M119872119873

(119892119909 119910 120601 (1199050)) M

119872119873(119892119910 119909 120601 (119905

0)))

ge119871lowastT ([M

119872119873(119892119909 119910 119905

0)]2

[M119872119873

(119892119910 119909 1199050)]2

) (55)


T (M119872119873

(119892119909 119910 120601119899

(1199050)) M

119872119873(119892119910 119909 120601

119899

(1199050)))

ge119871lowastT ([M

119872119873(119892119909 119910 120601

119899minus1

(1199050))]2

[M119872119873

(119892119910 119909 120601119899minus1

(1199050))]2

)

ge119871lowastT ([M

119872119873(119892119909 119910 119905

0)]2119899

[M119872119873

(119892119910 119909 1199050)]2119899

)

(56)


Thus

T (M119872119873

(119892119909 119910 120601119899

(1199050)) M

119872119873(119892119910 119909 120601

119899

(1199050)))

ge119871lowastT ([M

119872119873(119892119909 119910 119905

0)]2119899

[M119872119873

(119892119910 119909 1199050)]2119899

)

(57)


T (M119872119873

(119892119909 119910 119905) M119872119873

(119892119910 119909 119905))

ge119871lowastT(M

119872119873(119892119909 119910

infin

sum119896=1198990

120601119896

(1199050))

M119872119873

(119892119910 119909infin

sum119896=1198990

120601119896

(1199050)))

ge119871lowastT (M

119872119873(119892119909 119910 120601

1198990 (1199050)) M

119872119873(119892119910 119909 120601

1198990 (1199050)))

ge119871lowastT ([M

119872119873(119892119909 119910 119905

0)]21198990

[M119872119873

(119892119910 119909 1199050)]21198990

)

ge119871lowastT21198990+1

minus1

(119873119904(120583) 120583)

ge119871lowast (119873119904(120582) 120582)

(58)


T (M119872119873

(119892119909 119910 119905) M119872119873

(119892119910 119909 119905)) ge119871lowast (119873119904(120582) 120582)

forall119905 gt 0

(59)




T119898minus1

(119873119904(120583) 120583) ge



M119872119873



M119872119873

(119909 119910 1199050) ge119871lowast (119873119904(120583) 120583) (61)



0isin 119873

such that

119905 gtinfin

sum119896=1198990

120601119896

(1199050) (62)


M119872119873

(119892119909119899+1

119892119910119899+1

120601 (1199050))

= M119872119873

(119865 (119909119899 119910119899) 119865 (119910

119899 119909119899) 120601 (119905

0))

ge119871lowastT (M

119872119873(119892119909119899 119892119910119899 1199050) M119872119873

(119892119910119899 119892119909119899 1199050))

(63)

thus

M119872119873

(119892119909119899+1

119892119910119899+1

120601 (1199050))

ge119871lowastT (M

119872119873(119892119909119899 119892119910119899 1199050)

M119872119873

(119892119910119899 119892119909119899 1199050))

(64)


M119872119873

(119909 119910 120601 (1199050))

ge119871lowastT (M

119872119873(119909 119910 119905

0) M119872119873

(119910 119909 1199050))

(65)


M119872119873

(119909 119910 119905)

ge119871lowastM119872119873

(119909 119910infin

sum119896=1198990

120601119896

(1199050))

ge119871lowast M119872119873

(119909 119910 1206011198990 (1199050))

ge119871lowastT ([M

119872119873(119909 119910 119905

0)]21198990minus1

[M119872119873

(119910 119909 1199050)]21198990minus1

)

ge119871lowastT21198990+1

minus3

(119873119904(120583) 120583)

ge119871lowast (119873119904(120582) 120582)

(66)



Corollary 32 Let (119883M119872119873


M119872119873

(119865 (119909 119910) 119865 (119906 V) 120601 (119905))

ge119871lowastT (M

119872119873(119909 119906 119905) M

119872119873(119910 V 119905))

(67)



Corollary 33 Let (119883M119872119873




M119872119873

(119865 (119909 119910) 119865 (119906 V) 119896119905)

ge119871lowastT (M

119872119873(119892119909 119892119906 119905) M

119872119873(119892119910 119892V 119905))

(68)








1 1198871max119886

2 1198872) for all 119886 = (119886

1 1198862) and


M119872119873

(119909 119910 119905) = (119905

119905 +1003816100381610038161003816119909 minus 119910

1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119905 +1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)

forall119909 119910 isin 119883 119905 gt 0

(69)

Then (119883M119872119873


119892 (119909) =

0 119909 = 0

1 119909 =1

2119899 + 1

1

2119899 + 1 119909 =

1

2119899

119865 (119909 119910) =

1

(2119899 + 1)4 (119909 119910) = (

1

21198991

2119899)

0 otherwise

(70)

Let 119909119899= 119910119899= 12119899 We have

119892119909119899=

1

2119899 + 1997888rarr 0

119865 (119909119899 119910119899) =

1


(71)

but

M119872119873

(119865 (119892119909119899 119892119910119899) 119892119865 (119909

119899 119910119899) 119905)

= M119872119873

(0 1 119905) 999424999426999456 1119871lowast

(72)


119905

119883 + 119905ge min 119905

119884 + 119905

119905

119885 + 119905

119883

119883 + 119905le max 119884

119884 + 119905

119885

119885 + 119905


(73)



M119872119873

(119865 (119909 119910) 119865 (119906 V) 120601 (119905))

ge119871lowastT (M

119872119873(119892119909 119892119906 119905) M

119872119873(119892119910 119892V 119905))

(74)


21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)


1003816100381610038161003816 (75)





21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816 =2

(2119899 + 1)4

max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V

1003816100381610038161003816 =2119899

2119899 + 1

(76)




21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816 =2

(2119899 + 1)4

max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V

1003816100381610038161003816

= max 1

2119899 + 110038161003816100381610038161003816100381610038161003816

1

2119895 + 1minus

1

2119899 + 1

10038161003816100381610038161003816100381610038161003816

(77)

or

max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V

1003816100381610038161003816

= max 2119899

2119899 + 110038161003816100381610038161003816100381610038161003816

1

2119895 + 1minus

1

2119899 + 1

10038161003816100381610038161003816100381610038161003816

(78)

So (75) holds


21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816 =2

(2119899 + 1)4

max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V

1003816100381610038161003816

= max1003816100381610038161003816100381610038161003816

1

2119894 + 1minus

1

2119899 + 1

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1

2119895 + 1minus

1

2119899 + 1

10038161003816100381610038161003816100381610038161003816

(79)



21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816 = 2

100381610038161003816100381610038161003816100381610038161003816

1

(2119894 + 1)4minus

1

(2119899 + 1)4

100381610038161003816100381610038161003816100381610038161003816

max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V

1003816100381610038161003816

=1

2119894 + 1minus

1

2119899 + 1

(80)




119909 (119901) = int119887

119886

(1198701(119901 119902) + 119870

2(119901 119902))

times [119891 (119902 119909 (119902)) + 119892 (119902 119909 (119902))] 119889119902 + ℎ (119901)

(81)




(ii120579) 120579(119901) le 119901




1198701(119901 119902) ge 0 119870

2(119901 119902) le 0 forall119901 119902 isin 119868 (82)


0 le 119891 (119902 119909) minus 119891 (119902 119910) le 120582120579 (119909 minus 119910)


(iii) Consider

max 120582 120583 sup119901isin119868

int119887

119886

[1198701(119901 119902) minus 119870

2(119901 119902)] 119889119902 le

1

8 (84)






M119872119873

(119909 119910 119905) = (119905

119905 +1003816100381610038161003816119909 minus 119910

1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119905 +1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)

forall119909 119910 isin 119883 119905 gt 0

(85)


1 1198871max119886

2 1198872) for all 119886 = (119886

1 1198862) and 119887 = (119887

1 1198872) isin


119865 (119909 119910) (119901) = int119887

119886

1198701(119901 119902) [119891 (119902 119909 (119902)) + 119892 (119902 119910 (119902))] 119889119902

+ int119887

119886

1198702(119901 119902) [119891 (119902 119910 (119902))

+119892 (119902 119909 (119902))] 119889119902 + ℎ (119901)

(86)


119865 (119909 119910) (119901) minus 119865 (119906 V) (119901)

= int119887

119886

1198701(119901 119902) [119891 (119902 119909 (119902)) + 119892 (119902 119910 (119902))] 119889119902

+ int119887

119886

1198702(119901 119902) [119891 (119902 119910 (119902)) + 119892 (119902 119909 (119902))] 119889119902

minus int119887

119886

1198701(119901 119902) [119891 (119902 119906 (119902)) + 119892 (119902 V (119902))] 119889119902

minus int119887

119886

1198702(119901 119902) [119891 (119902 V (119902)) + 119892 (119902 119906 (119902))] 119889119902

= int119887

119886

1198701(119901 119902) [119891 (119902 119909 (119902)) minus 119891 (119902 119906 (119902))

+119892 (119902 119910 (119902)) minus 119892 (119902 V (119902))] 119889119902

+ int119887

119886

1198702(119901 119902) [119891 (119902 119910 (119902)) minus 119891 (119902 V (119902))

+119892 (119902 119909 (119902)) minus 119892 (119902 119906 (119902))] 119889119902

= int119887

119886

1198701(119901 119902) [(119891 (119902 119909 (119902)) minus 119891 (119902 119906 (119902)))

minus (119892 (119902 V (119902)) minus 119892 (119902 119910 (119902)))] 119889119902

minus int119887

119886

1198702(119901 119902) [(119891 (119902 V (119902)) minus 119891 (119902 119910 (119902)))

minus (119892 (119902 119909 (119902)) minus 119892 (119902 119906 (119902)))] 119889119902

le int119887

119886

1198701(119901 119902) [120582120579 (119909 (119902) minus 119906 (119902))

+120583120579 (V (119902) minus 119910 (119902))] 119889119902


minus int119887

119886

1198702(119901 119902) [120582120579 (V (119902) minus 119910 (119902))

+120583120579 (119909 (119902) minus 119906 (119902))] 119889119902

(87)

Thus

119865 (119909 119910) (119901) minus 119865 (119906 V) (119901)

le int119887

119886

1198701(119901 119902) [120582120579 (119909 (119902) minus 119906 (119902))

+120583120579 (V (119902) minus 119910 (119902))] 119889119902

minus int119887

119886

1198702(119901 119902) [120582120579 (V (119902) minus 119910 (119902))

+120583120579 (119909 (119902) minus 119906 (119902))] 119889119902

(88)


120579 (119909 (119902) minus 119906 (119902)) le 120579 (1003816100381610038161003816119909 (119902) minus 119906 (119902)

1003816100381610038161003816)

120579 (V (119902) minus 119910 (119902)) le 120579 (1003816100381610038161003816V (119902) minus 119910 (119902)

1003816100381610038161003816) (89)


1003816100381610038161003816119865 (119909 119910) (119901) minus 119865 (119906 V) (119901)1003816100381610038161003816

le int119887

119886

1198701(119901 119902) [120582120579 (

1003816100381610038161003816119909 (119902) minus 119906 (119902)1003816100381610038161003816)

+120583120579 (1003816100381610038161003816V (119902) minus 119910 (119902)

1003816100381610038161003816)] 119889119902

minus int119887

119886

1198702(119901 119902) [120582120579 (

1003816100381610038161003816V (119902) minus 119910 (119902)1003816100381610038161003816)

+120583120579 (1003816100381610038161003816119909 (119902) minus 119906 (119902)

1003816100381610038161003816)] 119889119902

le int119887

119886

1198701(119901 119902) [max 120582 120583 120579 (1003816100381610038161003816119909 (119902) minus 119906 (119902)

1003816100381610038161003816)

+max 120582 120583 120579 (1003816100381610038161003816V (119902) minus 119910 (119902)1003816100381610038161003816)] 119889119902

minus int119887

119886

1198702(119901 119902) [max 120582 120583 120579 (1003816100381610038161003816V (119902) minus 119910 (119902)

1003816100381610038161003816)

+max 120582 120583 120579 (1003816100381610038161003816119909 (119902) minus 119906 (119902)1003816100381610038161003816)] 119889119902

(90)


1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)1003816100381610038161003816

le max 120582 120583 int119887

119886

[1198701(119901 119902) minus 119870

2(119901 119902)] 119889119902

sdot [120579 (1003816100381610038161003816119909 (119902)minus119906 (119902)

1003816100381610038161003816)+120579 (1003816100381610038161003816V (119902) minus 119910 (119902)

1003816100381610038161003816)]

le max 120582 120583 sup119901isin119868

int119887

119886

[1198701(119901 119902) minus 119870

2(119901 119902)] 119889119902

sdot [120579 (1003816100381610038161003816119909 (119902)minus119906 (119902)

1003816100381610038161003816)+120579 (1003816100381610038161003816V (119902) minus 119910 (119902)

1003816100381610038161003816)]

le120579 (|119909 minus 119906|) + 120579 (

1003816100381610038161003816V minus 1199101003816100381610038161003816)

8

(91)

Thus

21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816 le120579 (|119909 minus 119906|) + 120579 (

1003816100381610038161003816V minus 1199101003816100381610038161003816)

4 (92)



120579 (1003816100381610038161003816119910 minus V1003816100381610038161003816) le 120579 (|119909 minus 119906| +

1003816100381610038161003816119910 minus V1003816100381610038161003816) (93)


120579 (|119909 minus 119906|) + 120579 (1003816100381610038161003816119910 minus V1003816100381610038161003816)

2le 120579 (|119909 minus 119906| +

1003816100381610038161003816119910 minus V1003816100381610038161003816)

le |119909 minus 119906| +1003816100381610038161003816119910 minus V1003816100381610038161003816


(94)

and so

120579 (|119909 minus 119906|) + 120579 (1003816100381610038161003816119910 minus V1003816100381610038161003816)



21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)



M119872119873

(119865 (119909 119910) 119865 (119906 V) 120601 (119905))

= M119872119873

(119865 (119909 119910) 119865 (119906 V) 119905

2)

= (1199052

1199052 +1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816

1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)1003816100381610038161003816

1199052 +1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816)

= (119905

119905 + 21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816

21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816119905 + 2

1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)1003816100381610038161003816)

ge119871lowast (

119905

119905 +max |119909 minus 119906| 1003816100381610038161003816119910 minus V1003816100381610038161003816


)



119905 + |119909 minus 119906|

119905

119905 +1003816100381610038161003816119910 minus V1003816100381610038161003816

max |119909 minus 119906|

119905 + |119909 minus 119906|

1003816100381610038161003816119910 minus V1003816100381610038161003816119905 +

1003816100381610038161003816119910 minus V1003816100381610038161003816)

ge119871lowastT (M

119872119873(119909 119906 119905) M

119872119873(119910 V 119905))

(97)




References




























Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in





defined as follows

M119872119873

(119909 119910 119905) = (119872 (119909 119910 119905) 119873 (119909 119910 119905))

=

(119909

119910119910 minus 119909

119910) if 119909 le 119910

(119910

119909119909 minus 119910

119909) if 119910 le 119909

(6)







119871lowast







M119872119873

(119909119899 119909119898 119905) gt119871lowast (119873119904(120576) 120576) (7)





T) and

denoted by 119909119899

M119872119873

997888997888997888997888rarr 119909 if M119872119873

(119909119899 119909 119905) rarr 1





119872119873(119909 119910 119905)




119861 (119909 119903 119905) = 119910 isin 119883 M119872119873

(119909 119910 119905) gt119871lowast (119873119904(119903) 119903) (8)






lim119899rarrinfin

M119872119873

(119909119899 119910119899 119905119899) = M

119872119873(119909 119910 119905) (9)



lim119899rarrinfin

M119872119873

(119909119899 119910 119905) = M

119872119873(119909 119910 119905)

lim119899rarrinfin

M119872119873

(119909 119910119899 119905) = M

119872119873(119909 119910 119905)

lim119899rarrinfin

M119872119873

(119909 119910 119905119899) = M

119872119873(119909 119910 119905)

(10)




119865 (119909 119910) = 119909 119865 (119910 119909) = 119910 (11)


119865 (119909 119910) = 119892 (119909) 119865 (119910 119909) = 119892 (119910) (12)


119909 = 119865 (119909 119910) = 119892 (119909) 119910 = 119865 (119910 119909) = 119892 (119910) (13)


119909 = 119865 (119909 119909) = 119892 (119909) (14)





lim119899rarrinfin

M119872119873

(119892119865 (119909119899 119910119899) 119865 (119892119909

119899 119892119910119899) 119905) = 1

119871lowast

lim119899rarrinfin

M119872119873

(119892119865 (119910119899 119909119899) 119865 (119892119910

119899 119892119909119899) 119905) = 1

119871lowast

(16)




119865 (119909119899 119910119899) = lim119899rarrinfin

119892 (119909119899) = 119909

lim119899rarrinfin

119865 (119910119899 119909119899) = lim119899rarrinfin

119892 (119910119899) = 119910 forall119909 119910 isin 119883

(17)



3 Main Results






T1

(119905) = T (119905 119905) T119898+1

(119905) = T (119905T119898

(119905))

119898 = 1 2 119905 isin 119871lowast

(18)


T119898


119871lowast (119873119904(120583) 120583)

(19)



119872119873(119909 119910 119905)gt

119871lowast(119873119904(119903) 119903) for


0lt 119905

such that M119872119873

(119909 119910 1199050) gt (119873

119904(119903) 119903)


[M119872119873

(119909 119910 119905)]119899

= T119899minus1

(M119872119873

(119909 119910 119905)) (20)

for all 119899 isin 119873

Definition 26 Let (119883M119872119873




lim119901rarrinfin

[M119872119873

(119909 119910 119896119901

119905)]119901

= 1119871lowast (21)


Lemma 27 Let (119883M119872119873



lim119905rarrinfin

M119872119873





lim119905rarrinfin

M119872119873

(1199090 1199100 119905) = (120582119873

119904(120582)) lt 1

119871lowast (23)


lim119901rarrinfin

[M119872119873

(1199090 1199100 119896119901

119905)]119901

= 0119871lowast (24)




120595 (119905) =

120572radic119905 0 le 119905 le 4

1 minus1

ln 119905 119905 gt 4

(25)


119905rarrinfin120595(119905) = 1 Let

M119872119873

(119909 119910 119905) = ([120595 (119905)]119889(119909119910)

1 minus [120595 (119905)]119889(119909119910)

) (26)


for all 119886 = (1198861 1198862) and 119887 = (119887


119872119873T)


lim119905rarrinfin

M119872119873

(119909 119910 119905)


([120595 (119905)]119889(119909119910)

1 minus [120595 (119905)]119889(119909119910)

)


(27)


lim119901rarrinfin

[M119872119873

(119909 119910 119896119901

119905)]119901


[120595 (119896119901

119905)]119889(119909119910)sdot119901

lim119901rarrinfin

[1 minus [120595 (119896119901

119905)]119889(119909119910)sdot119901

])


[1 minus1

ln (119896119901119905)]119889(119909119910)sdot119901

lim119901rarrinfin

[1 minus [1 minus1

ln(119896119901119905)]119901sdot119889(119909119910)

])

= (119890minus119889(119909119910) ln 119896

1 minus 119890minus119889(119909119910) ln 119896

)

= 1119871lowast

(28)




120601119899(120601(119905)) 119899 isin 119873


Lemma 30 Let (119883M119872119873


M119872119873

(119909 119910 120601 (119905)) ge119871lowastM119872119873

(119909 119910 119905) forall119905 gt 0 (29)

then 119909 = 119910


we have M119872119873

(119909 119910 119905)ge119871lowastM119872119873


Theorem 31 Let (119883M119872119873




M119872119873

(119865 (119909 119910) 119865 (119906 V) 120601 (119905))

ge119871lowastT (M

119872119873(119892119909 119892119906 119905) M

119872119873(119892119910 119892V 119905))

(30)




1198921199091= 119865 (119909

0 1199100) 119892119910

1= 119865 (119910

0 1199090) (31)


and 119910119899 in119883 such that

119892119909119899+1

= 119865 (119909119899 119910119899) 119892119910

119899+1= 119865 (119910

119899 119909119899) forall119899 ge 0 (32)





T119898minus1

(119873119904(120583) 120583) ge



M119872119873



M119872119873

(1198921199090 1198921199091 1199050) ge119871lowast (119873119904(120583) 120583)

M119872119873

(1198921199100 1198921199101 1199050) ge119871lowast (119873119904(120583) 120583)

(34)



0isin 119873

such that

119905 gtinfin

sum119896=1198990

120601119896

(1199050) (35)


M119872119873

(1198921199091 1198921199092 120601 (1199050))

= M119872119873

(119865 (1199090 1199100) 119865 (119909

1 1199101) 120601 (119905

0))

ge119871lowastT (M

119872119873(1198921199090 1198921199091 1199050) M119872119873

(1198921199100 1198921199101 1199050))

M119872119873

(1198921199101 1198921199102 120601 (1199050))

= M119872119873

(119865 (1199100 1199090) 119865 (119910

1 1199091) 120601 (119905

0))

ge119871lowastT (M

119872119873(1198921199100 1198921199101 1199050) M119872119873

(1198921199090 1198921199091 1199050))

(36)


M119872119873

(1198921199092 1198921199093 1206012

(1199050))

= M119872119873

(119865 (1199091 1199101) 119865 (119909

2 1199102) 1206012

(1199050))

ge119871lowastT (M

119872119873(1198921199091 1198921199092 120601 (1199050))

M119872119873

(1198921199101 1198921199102 120601 (1199050)))

ge119871lowastT ([M

119872119873(1198921199090 1198921199091 1199050)]2

[M119872119873

(1198921199100 1198921199101 1199050)]2

)

M119872119873

(1198921199102 1198921199103 1206012

(1199050))

= M119872119873

(119865 (1199101 1199091) 119865 (119910

2 1199092) 1206012

(1199050))

ge119871lowastT (M

119872119873(1198921199101 1198921199102 120601 (1199050))

M119872119873

(1198921199091 1198921199092 120601 (1199050)))

ge119871lowastT ([M

119872119873(1198921199100 1198921199101 1199050)]2

[M119872119873

(1198921199090 1198921199091 1199050)]2

)

(37)


M119872119873

(119892119909119899 119892119909119899+1

120601119899

(1199050))

ge119871lowastT ([M

119872119873(1198921199090 1198921199091 1199050)]2119899minus1

[M119872119873

(1198921199100 1198921199101 1199050)]2119899minus1

)

M119872119873

(119892119910119899 119892119910119899+1

120601119899

(1199050))

ge119871lowastT ([M

119872119873(1198921199100 1198921199101 1199050)]2119899minus1

[M119872119873

(1198921199090 1198921199091 1199050)]2119899minus1

)

(38)


M119872119873

(119892119909119899 119892119909119898 119905)

ge119871lowastM119872119873

(119892119909119899 119892119909119898infin

sum119896=1198990

120601119896

(1199050))

ge119871lowastM119872119873

(119892119909119899 119892119909119898119898minus1

sum119896=119899

120601119896

(1199050))


(M119872119873

(119892119909119899 119892119909119899+1

120601119899

(1199050))

M119872119873

(119892119909119899+1

119892119909119899+2

120601119899+1

(1199050))

M119872119873

(119892119909119898minus1

119892119909119898 120601119898minus1

(1199050)))


(T ([M119872119873

(1198921199090 1198921199091 1199050)]2119899minus1

[M119872119873

(1198921199100 1198921199101 1199050)]2119899minus1

)


T ([M119872119873

(1198921199090 1198921199091 1199050)]2119899

[M119872119873

(1198921199100 1198921199101 1199050)]2119899

)

T ([M119872119873

(1198921199090 1198921199091 1199050)]2119898minus2

[M119872119873

(1198921199100 1198921199101 1199050)]2119898minus2

))

ge119871lowastT ([M

119872119873(1198921199090 1198921199091 1199050)]2119898minus1

minus2119899minus1

[M119872119873

(1198921199100 1198921199101 1199050)]2119898minus1

minus2119899minus1

)


minus2119899

minus1

(119873119904(120583) 120583)

ge119871lowast (119873119904(120582) 120582)

(39)

which implies that

M119872119873

(119892119909119899 119892119909119898 119905) gt119871lowast (119873119904(120582) 120582) (40)


119899 is


Cauchy sequence



lim119899rarrinfin

119892119909119899= lim119899rarrinfin

119865 (119909119899 119910119899) = 119892119886 = 119909

lim119899rarrinfin

119892119910119899= lim119899rarrinfin

119865 (119910119899 119909119899) = 119892119887 = 119910

(41)

From (30) we get

M119872119873

(119865 (119909119899 119910119899) 119865 (119886 119887) 120601 (119905))

ge119871lowastT (M

119872119873(119892119909119899 119892119886 119905) M

119872119873(119892119910119899 119892119887 119905))

(42)

SinceM119872119873


M119872119873

(119892119886 119865 (119886 119887) 120601 (119905)) = 1119871lowast (43)

which implies that

119865 (119886 119887) = 119892119886 = 119909 (44)


119865 (119887 119886) = 119892119887 = 119910 (45)


119892119865 (119886 119887) = 119865 (119892119886 119892119887) 119892119865 (119887 119886) = 119865 (119892119887 119892119886) (46)

which implies that

119892119909 = 119865 (119909 119910) 119892119910 = 119865 (119910 119909) (47)



T119898minus1

(119873119904(120583) 120583) ge



M119872119873



M119872119873

(119892119909 119910 1199050) ge119871lowast (119873119904(120583) 120583)

M119872119873

(119892119910 119909 1199050) ge119871lowast (119873119904(120583) 120583)

(49)



0isin 119873

such that

119905 gtinfin

sum119896=1198990

120601119896

(1199050) (50)

Since

M119872119873

(119892119909 119892119910119899+1

120601 (1199050))

= M119872119873

(119865 (119909 119910) 119865 (119910119899 119909119899) 120601 (119905

0))

ge119871lowastT (M

119872119873(119892119909 119892119910

119899 1199050) lowastM

119872119873(119892119910 119892119909

119899 1199050))

(51)

thus

M119872119873

(119892119909 119892119910119899+1

120601 (1199050))

ge119871lowastT (M

119872119873(119892119909 119892119910

119899 1199050) lowastM

119872119873(119892119910 119892119909

119899 1199050))

(52)


M119872119873

(119892119909 119910 120601 (1199050))

ge119871lowastT (M

119872119873(119892119909 119910 119905

0) M119872119873

(119892119910 119909 1199050))

(53)


M119872119873

(119892119910 119909 120601 (1199050))

ge119871lowastT (M

119872119873(119892119909 119910 119905

0) M119872119873

(119892119910 119909 1199050))

(54)


T (M119872119873

(119892119909 119910 120601 (1199050)) M

119872119873(119892119910 119909 120601 (119905

0)))

ge119871lowastT ([M

119872119873(119892119909 119910 119905

0)]2

[M119872119873

(119892119910 119909 1199050)]2

) (55)


T (M119872119873

(119892119909 119910 120601119899

(1199050)) M

119872119873(119892119910 119909 120601

119899

(1199050)))

ge119871lowastT ([M

119872119873(119892119909 119910 120601

119899minus1

(1199050))]2

[M119872119873

(119892119910 119909 120601119899minus1

(1199050))]2

)

ge119871lowastT ([M

119872119873(119892119909 119910 119905

0)]2119899

[M119872119873

(119892119910 119909 1199050)]2119899

)

(56)


Thus

T (M119872119873

(119892119909 119910 120601119899

(1199050)) M

119872119873(119892119910 119909 120601

119899

(1199050)))

ge119871lowastT ([M

119872119873(119892119909 119910 119905

0)]2119899

[M119872119873

(119892119910 119909 1199050)]2119899

)

(57)


T (M119872119873

(119892119909 119910 119905) M119872119873

(119892119910 119909 119905))

ge119871lowastT(M

119872119873(119892119909 119910

infin

sum119896=1198990

120601119896

(1199050))

M119872119873

(119892119910 119909infin

sum119896=1198990

120601119896

(1199050)))

ge119871lowastT (M

119872119873(119892119909 119910 120601

1198990 (1199050)) M

119872119873(119892119910 119909 120601

1198990 (1199050)))

ge119871lowastT ([M

119872119873(119892119909 119910 119905

0)]21198990

[M119872119873

(119892119910 119909 1199050)]21198990

)

ge119871lowastT21198990+1

minus1

(119873119904(120583) 120583)

ge119871lowast (119873119904(120582) 120582)

(58)


T (M119872119873

(119892119909 119910 119905) M119872119873

(119892119910 119909 119905)) ge119871lowast (119873119904(120582) 120582)

forall119905 gt 0

(59)




T119898minus1

(119873119904(120583) 120583) ge



M119872119873



M119872119873

(119909 119910 1199050) ge119871lowast (119873119904(120583) 120583) (61)



0isin 119873

such that

119905 gtinfin

sum119896=1198990

120601119896

(1199050) (62)


M119872119873

(119892119909119899+1

119892119910119899+1

120601 (1199050))

= M119872119873

(119865 (119909119899 119910119899) 119865 (119910

119899 119909119899) 120601 (119905

0))

ge119871lowastT (M

119872119873(119892119909119899 119892119910119899 1199050) M119872119873

(119892119910119899 119892119909119899 1199050))

(63)

thus

M119872119873

(119892119909119899+1

119892119910119899+1

120601 (1199050))

ge119871lowastT (M

119872119873(119892119909119899 119892119910119899 1199050)

M119872119873

(119892119910119899 119892119909119899 1199050))

(64)


M119872119873

(119909 119910 120601 (1199050))

ge119871lowastT (M

119872119873(119909 119910 119905

0) M119872119873

(119910 119909 1199050))

(65)


M119872119873

(119909 119910 119905)

ge119871lowastM119872119873

(119909 119910infin

sum119896=1198990

120601119896

(1199050))

ge119871lowast M119872119873

(119909 119910 1206011198990 (1199050))

ge119871lowastT ([M

119872119873(119909 119910 119905

0)]21198990minus1

[M119872119873

(119910 119909 1199050)]21198990minus1

)

ge119871lowastT21198990+1

minus3

(119873119904(120583) 120583)

ge119871lowast (119873119904(120582) 120582)

(66)



Corollary 32 Let (119883M119872119873


M119872119873

(119865 (119909 119910) 119865 (119906 V) 120601 (119905))

ge119871lowastT (M

119872119873(119909 119906 119905) M

119872119873(119910 V 119905))

(67)



Corollary 33 Let (119883M119872119873




M119872119873

(119865 (119909 119910) 119865 (119906 V) 119896119905)

ge119871lowastT (M

119872119873(119892119909 119892119906 119905) M

119872119873(119892119910 119892V 119905))

(68)








1 1198871max119886

2 1198872) for all 119886 = (119886

1 1198862) and


M119872119873

(119909 119910 119905) = (119905

119905 +1003816100381610038161003816119909 minus 119910

1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119905 +1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)

forall119909 119910 isin 119883 119905 gt 0

(69)

Then (119883M119872119873


119892 (119909) =

0 119909 = 0

1 119909 =1

2119899 + 1

1

2119899 + 1 119909 =

1

2119899

119865 (119909 119910) =

1

(2119899 + 1)4 (119909 119910) = (

1

21198991

2119899)

0 otherwise

(70)

Let 119909119899= 119910119899= 12119899 We have

119892119909119899=

1

2119899 + 1997888rarr 0

119865 (119909119899 119910119899) =

1


(71)

but

M119872119873

(119865 (119892119909119899 119892119910119899) 119892119865 (119909

119899 119910119899) 119905)

= M119872119873

(0 1 119905) 999424999426999456 1119871lowast

(72)


119905

119883 + 119905ge min 119905

119884 + 119905

119905

119885 + 119905

119883

119883 + 119905le max 119884

119884 + 119905

119885

119885 + 119905


(73)



M119872119873

(119865 (119909 119910) 119865 (119906 V) 120601 (119905))

ge119871lowastT (M

119872119873(119892119909 119892119906 119905) M

119872119873(119892119910 119892V 119905))

(74)


21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)


1003816100381610038161003816 (75)





21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816 =2

(2119899 + 1)4

max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V

1003816100381610038161003816 =2119899

2119899 + 1

(76)




21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816 =2

(2119899 + 1)4

max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V

1003816100381610038161003816

= max 1

2119899 + 110038161003816100381610038161003816100381610038161003816

1

2119895 + 1minus

1

2119899 + 1

10038161003816100381610038161003816100381610038161003816

(77)

or

max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V

1003816100381610038161003816

= max 2119899

2119899 + 110038161003816100381610038161003816100381610038161003816

1

2119895 + 1minus

1

2119899 + 1

10038161003816100381610038161003816100381610038161003816

(78)

So (75) holds


21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816 =2

(2119899 + 1)4

max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V

1003816100381610038161003816

= max1003816100381610038161003816100381610038161003816

1

2119894 + 1minus

1

2119899 + 1

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1

2119895 + 1minus

1

2119899 + 1

10038161003816100381610038161003816100381610038161003816

(79)



21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816 = 2

100381610038161003816100381610038161003816100381610038161003816

1

(2119894 + 1)4minus

1

(2119899 + 1)4

100381610038161003816100381610038161003816100381610038161003816

max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V

1003816100381610038161003816

=1

2119894 + 1minus

1

2119899 + 1

(80)




119909 (119901) = int119887

119886

(1198701(119901 119902) + 119870

2(119901 119902))

times [119891 (119902 119909 (119902)) + 119892 (119902 119909 (119902))] 119889119902 + ℎ (119901)

(81)




(ii120579) 120579(119901) le 119901




1198701(119901 119902) ge 0 119870

2(119901 119902) le 0 forall119901 119902 isin 119868 (82)


0 le 119891 (119902 119909) minus 119891 (119902 119910) le 120582120579 (119909 minus 119910)


(iii) Consider

max 120582 120583 sup119901isin119868

int119887

119886

[1198701(119901 119902) minus 119870

2(119901 119902)] 119889119902 le

1

8 (84)






M119872119873

(119909 119910 119905) = (119905

119905 +1003816100381610038161003816119909 minus 119910

1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119905 +1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)

forall119909 119910 isin 119883 119905 gt 0

(85)


1 1198871max119886

2 1198872) for all 119886 = (119886

1 1198862) and 119887 = (119887

1 1198872) isin


119865 (119909 119910) (119901) = int119887

119886

1198701(119901 119902) [119891 (119902 119909 (119902)) + 119892 (119902 119910 (119902))] 119889119902

+ int119887

119886

1198702(119901 119902) [119891 (119902 119910 (119902))

+119892 (119902 119909 (119902))] 119889119902 + ℎ (119901)

(86)


119865 (119909 119910) (119901) minus 119865 (119906 V) (119901)

= int119887

119886

1198701(119901 119902) [119891 (119902 119909 (119902)) + 119892 (119902 119910 (119902))] 119889119902

+ int119887

119886

1198702(119901 119902) [119891 (119902 119910 (119902)) + 119892 (119902 119909 (119902))] 119889119902

minus int119887

119886

1198701(119901 119902) [119891 (119902 119906 (119902)) + 119892 (119902 V (119902))] 119889119902

minus int119887

119886

1198702(119901 119902) [119891 (119902 V (119902)) + 119892 (119902 119906 (119902))] 119889119902

= int119887

119886

1198701(119901 119902) [119891 (119902 119909 (119902)) minus 119891 (119902 119906 (119902))

+119892 (119902 119910 (119902)) minus 119892 (119902 V (119902))] 119889119902

+ int119887

119886

1198702(119901 119902) [119891 (119902 119910 (119902)) minus 119891 (119902 V (119902))

+119892 (119902 119909 (119902)) minus 119892 (119902 119906 (119902))] 119889119902

= int119887

119886

1198701(119901 119902) [(119891 (119902 119909 (119902)) minus 119891 (119902 119906 (119902)))

minus (119892 (119902 V (119902)) minus 119892 (119902 119910 (119902)))] 119889119902

minus int119887

119886

1198702(119901 119902) [(119891 (119902 V (119902)) minus 119891 (119902 119910 (119902)))

minus (119892 (119902 119909 (119902)) minus 119892 (119902 119906 (119902)))] 119889119902

le int119887

119886

1198701(119901 119902) [120582120579 (119909 (119902) minus 119906 (119902))

+120583120579 (V (119902) minus 119910 (119902))] 119889119902


minus int119887

119886

1198702(119901 119902) [120582120579 (V (119902) minus 119910 (119902))

+120583120579 (119909 (119902) minus 119906 (119902))] 119889119902

(87)

Thus

119865 (119909 119910) (119901) minus 119865 (119906 V) (119901)

le int119887

119886

1198701(119901 119902) [120582120579 (119909 (119902) minus 119906 (119902))

+120583120579 (V (119902) minus 119910 (119902))] 119889119902

minus int119887

119886

1198702(119901 119902) [120582120579 (V (119902) minus 119910 (119902))

+120583120579 (119909 (119902) minus 119906 (119902))] 119889119902

(88)


120579 (119909 (119902) minus 119906 (119902)) le 120579 (1003816100381610038161003816119909 (119902) minus 119906 (119902)

1003816100381610038161003816)

120579 (V (119902) minus 119910 (119902)) le 120579 (1003816100381610038161003816V (119902) minus 119910 (119902)

1003816100381610038161003816) (89)


1003816100381610038161003816119865 (119909 119910) (119901) minus 119865 (119906 V) (119901)1003816100381610038161003816

le int119887

119886

1198701(119901 119902) [120582120579 (

1003816100381610038161003816119909 (119902) minus 119906 (119902)1003816100381610038161003816)

+120583120579 (1003816100381610038161003816V (119902) minus 119910 (119902)

1003816100381610038161003816)] 119889119902

minus int119887

119886

1198702(119901 119902) [120582120579 (

1003816100381610038161003816V (119902) minus 119910 (119902)1003816100381610038161003816)

+120583120579 (1003816100381610038161003816119909 (119902) minus 119906 (119902)

1003816100381610038161003816)] 119889119902

le int119887

119886

1198701(119901 119902) [max 120582 120583 120579 (1003816100381610038161003816119909 (119902) minus 119906 (119902)

1003816100381610038161003816)

+max 120582 120583 120579 (1003816100381610038161003816V (119902) minus 119910 (119902)1003816100381610038161003816)] 119889119902

minus int119887

119886

1198702(119901 119902) [max 120582 120583 120579 (1003816100381610038161003816V (119902) minus 119910 (119902)

1003816100381610038161003816)

+max 120582 120583 120579 (1003816100381610038161003816119909 (119902) minus 119906 (119902)1003816100381610038161003816)] 119889119902

(90)


1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)1003816100381610038161003816

le max 120582 120583 int119887

119886

[1198701(119901 119902) minus 119870

2(119901 119902)] 119889119902

sdot [120579 (1003816100381610038161003816119909 (119902)minus119906 (119902)

1003816100381610038161003816)+120579 (1003816100381610038161003816V (119902) minus 119910 (119902)

1003816100381610038161003816)]

le max 120582 120583 sup119901isin119868

int119887

119886

[1198701(119901 119902) minus 119870

2(119901 119902)] 119889119902

sdot [120579 (1003816100381610038161003816119909 (119902)minus119906 (119902)

1003816100381610038161003816)+120579 (1003816100381610038161003816V (119902) minus 119910 (119902)

1003816100381610038161003816)]

le120579 (|119909 minus 119906|) + 120579 (

1003816100381610038161003816V minus 1199101003816100381610038161003816)

8

(91)

Thus

21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816 le120579 (|119909 minus 119906|) + 120579 (

1003816100381610038161003816V minus 1199101003816100381610038161003816)

4 (92)



120579 (1003816100381610038161003816119910 minus V1003816100381610038161003816) le 120579 (|119909 minus 119906| +

1003816100381610038161003816119910 minus V1003816100381610038161003816) (93)


120579 (|119909 minus 119906|) + 120579 (1003816100381610038161003816119910 minus V1003816100381610038161003816)

2le 120579 (|119909 minus 119906| +

1003816100381610038161003816119910 minus V1003816100381610038161003816)

le |119909 minus 119906| +1003816100381610038161003816119910 minus V1003816100381610038161003816


(94)

and so

120579 (|119909 minus 119906|) + 120579 (1003816100381610038161003816119910 minus V1003816100381610038161003816)



21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)



M119872119873

(119865 (119909 119910) 119865 (119906 V) 120601 (119905))

= M119872119873

(119865 (119909 119910) 119865 (119906 V) 119905

2)

= (1199052

1199052 +1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816

1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)1003816100381610038161003816

1199052 +1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816)

= (119905

119905 + 21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816

21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816119905 + 2

1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)1003816100381610038161003816)

ge119871lowast (

119905

119905 +max |119909 minus 119906| 1003816100381610038161003816119910 minus V1003816100381610038161003816


)



119905 + |119909 minus 119906|

119905

119905 +1003816100381610038161003816119910 minus V1003816100381610038161003816

max |119909 minus 119906|

119905 + |119909 minus 119906|

1003816100381610038161003816119910 minus V1003816100381610038161003816119905 +

1003816100381610038161003816119910 minus V1003816100381610038161003816)

ge119871lowastT (M

119872119873(119909 119906 119905) M

119872119873(119910 V 119905))

(97)




References




























Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




3 Main Results






T1

(119905) = T (119905 119905) T119898+1

(119905) = T (119905T119898

(119905))

119898 = 1 2 119905 isin 119871lowast

(18)


T119898


119871lowast (119873119904(120583) 120583)

(19)



119872119873(119909 119910 119905)gt

119871lowast(119873119904(119903) 119903) for


0lt 119905

such that M119872119873

(119909 119910 1199050) gt (119873

119904(119903) 119903)


[M119872119873

(119909 119910 119905)]119899

= T119899minus1

(M119872119873

(119909 119910 119905)) (20)

for all 119899 isin 119873

Definition 26 Let (119883M119872119873




lim119901rarrinfin

[M119872119873

(119909 119910 119896119901

119905)]119901

= 1119871lowast (21)


Lemma 27 Let (119883M119872119873



lim119905rarrinfin

M119872119873





lim119905rarrinfin

M119872119873

(1199090 1199100 119905) = (120582119873

119904(120582)) lt 1

119871lowast (23)


lim119901rarrinfin

[M119872119873

(1199090 1199100 119896119901

119905)]119901

= 0119871lowast (24)




120595 (119905) =

120572radic119905 0 le 119905 le 4

1 minus1

ln 119905 119905 gt 4

(25)


119905rarrinfin120595(119905) = 1 Let

M119872119873

(119909 119910 119905) = ([120595 (119905)]119889(119909119910)

1 minus [120595 (119905)]119889(119909119910)

) (26)


for all 119886 = (1198861 1198862) and 119887 = (119887


119872119873T)


lim119905rarrinfin

M119872119873

(119909 119910 119905)


([120595 (119905)]119889(119909119910)

1 minus [120595 (119905)]119889(119909119910)

)


(27)


lim119901rarrinfin

[M119872119873

(119909 119910 119896119901

119905)]119901


[120595 (119896119901

119905)]119889(119909119910)sdot119901

lim119901rarrinfin

[1 minus [120595 (119896119901

119905)]119889(119909119910)sdot119901

])


[1 minus1

ln (119896119901119905)]119889(119909119910)sdot119901

lim119901rarrinfin

[1 minus [1 minus1

ln(119896119901119905)]119901sdot119889(119909119910)

])

= (119890minus119889(119909119910) ln 119896

1 minus 119890minus119889(119909119910) ln 119896

)

= 1119871lowast

(28)




120601119899(120601(119905)) 119899 isin 119873


Lemma 30 Let (119883M119872119873


M119872119873

(119909 119910 120601 (119905)) ge119871lowastM119872119873

(119909 119910 119905) forall119905 gt 0 (29)

then 119909 = 119910


we have M119872119873

(119909 119910 119905)ge119871lowastM119872119873


Theorem 31 Let (119883M119872119873




M119872119873

(119865 (119909 119910) 119865 (119906 V) 120601 (119905))

ge119871lowastT (M

119872119873(119892119909 119892119906 119905) M

119872119873(119892119910 119892V 119905))

(30)




1198921199091= 119865 (119909

0 1199100) 119892119910

1= 119865 (119910

0 1199090) (31)


and 119910119899 in119883 such that

119892119909119899+1

= 119865 (119909119899 119910119899) 119892119910

119899+1= 119865 (119910

119899 119909119899) forall119899 ge 0 (32)





T119898minus1

(119873119904(120583) 120583) ge



M119872119873



M119872119873

(1198921199090 1198921199091 1199050) ge119871lowast (119873119904(120583) 120583)

M119872119873

(1198921199100 1198921199101 1199050) ge119871lowast (119873119904(120583) 120583)

(34)



0isin 119873

such that

119905 gtinfin

sum119896=1198990

120601119896

(1199050) (35)


M119872119873

(1198921199091 1198921199092 120601 (1199050))

= M119872119873

(119865 (1199090 1199100) 119865 (119909

1 1199101) 120601 (119905

0))

ge119871lowastT (M

119872119873(1198921199090 1198921199091 1199050) M119872119873

(1198921199100 1198921199101 1199050))

M119872119873

(1198921199101 1198921199102 120601 (1199050))

= M119872119873

(119865 (1199100 1199090) 119865 (119910

1 1199091) 120601 (119905

0))

ge119871lowastT (M

119872119873(1198921199100 1198921199101 1199050) M119872119873

(1198921199090 1198921199091 1199050))

(36)


M119872119873

(1198921199092 1198921199093 1206012

(1199050))

= M119872119873

(119865 (1199091 1199101) 119865 (119909

2 1199102) 1206012

(1199050))

ge119871lowastT (M

119872119873(1198921199091 1198921199092 120601 (1199050))

M119872119873

(1198921199101 1198921199102 120601 (1199050)))

ge119871lowastT ([M

119872119873(1198921199090 1198921199091 1199050)]2

[M119872119873

(1198921199100 1198921199101 1199050)]2

)

M119872119873

(1198921199102 1198921199103 1206012

(1199050))

= M119872119873

(119865 (1199101 1199091) 119865 (119910

2 1199092) 1206012

(1199050))

ge119871lowastT (M

119872119873(1198921199101 1198921199102 120601 (1199050))

M119872119873

(1198921199091 1198921199092 120601 (1199050)))

ge119871lowastT ([M

119872119873(1198921199100 1198921199101 1199050)]2

[M119872119873

(1198921199090 1198921199091 1199050)]2

)

(37)


M119872119873

(119892119909119899 119892119909119899+1

120601119899

(1199050))

ge119871lowastT ([M

119872119873(1198921199090 1198921199091 1199050)]2119899minus1

[M119872119873

(1198921199100 1198921199101 1199050)]2119899minus1

)

M119872119873

(119892119910119899 119892119910119899+1

120601119899

(1199050))

ge119871lowastT ([M

119872119873(1198921199100 1198921199101 1199050)]2119899minus1

[M119872119873

(1198921199090 1198921199091 1199050)]2119899minus1

)

(38)


M119872119873

(119892119909119899 119892119909119898 119905)

ge119871lowastM119872119873

(119892119909119899 119892119909119898infin

sum119896=1198990

120601119896

(1199050))

ge119871lowastM119872119873

(119892119909119899 119892119909119898119898minus1

sum119896=119899

120601119896

(1199050))


(M119872119873

(119892119909119899 119892119909119899+1

120601119899

(1199050))

M119872119873

(119892119909119899+1

119892119909119899+2

120601119899+1

(1199050))

M119872119873

(119892119909119898minus1

119892119909119898 120601119898minus1

(1199050)))


(T ([M119872119873

(1198921199090 1198921199091 1199050)]2119899minus1

[M119872119873

(1198921199100 1198921199101 1199050)]2119899minus1

)


T ([M119872119873

(1198921199090 1198921199091 1199050)]2119899

[M119872119873

(1198921199100 1198921199101 1199050)]2119899

)

T ([M119872119873

(1198921199090 1198921199091 1199050)]2119898minus2

[M119872119873

(1198921199100 1198921199101 1199050)]2119898minus2

))

ge119871lowastT ([M

119872119873(1198921199090 1198921199091 1199050)]2119898minus1

minus2119899minus1

[M119872119873

(1198921199100 1198921199101 1199050)]2119898minus1

minus2119899minus1

)


minus2119899

minus1

(119873119904(120583) 120583)

ge119871lowast (119873119904(120582) 120582)

(39)

which implies that

M119872119873

(119892119909119899 119892119909119898 119905) gt119871lowast (119873119904(120582) 120582) (40)


119899 is


Cauchy sequence



lim119899rarrinfin

119892119909119899= lim119899rarrinfin

119865 (119909119899 119910119899) = 119892119886 = 119909

lim119899rarrinfin

119892119910119899= lim119899rarrinfin

119865 (119910119899 119909119899) = 119892119887 = 119910

(41)

From (30) we get

M119872119873

(119865 (119909119899 119910119899) 119865 (119886 119887) 120601 (119905))

ge119871lowastT (M

119872119873(119892119909119899 119892119886 119905) M

119872119873(119892119910119899 119892119887 119905))

(42)

SinceM119872119873


M119872119873

(119892119886 119865 (119886 119887) 120601 (119905)) = 1119871lowast (43)

which implies that

119865 (119886 119887) = 119892119886 = 119909 (44)


119865 (119887 119886) = 119892119887 = 119910 (45)


119892119865 (119886 119887) = 119865 (119892119886 119892119887) 119892119865 (119887 119886) = 119865 (119892119887 119892119886) (46)

which implies that

119892119909 = 119865 (119909 119910) 119892119910 = 119865 (119910 119909) (47)



T119898minus1

(119873119904(120583) 120583) ge



M119872119873



M119872119873

(119892119909 119910 1199050) ge119871lowast (119873119904(120583) 120583)

M119872119873

(119892119910 119909 1199050) ge119871lowast (119873119904(120583) 120583)

(49)



0isin 119873

such that

119905 gtinfin

sum119896=1198990

120601119896

(1199050) (50)

Since

M119872119873

(119892119909 119892119910119899+1

120601 (1199050))

= M119872119873

(119865 (119909 119910) 119865 (119910119899 119909119899) 120601 (119905

0))

ge119871lowastT (M

119872119873(119892119909 119892119910

119899 1199050) lowastM

119872119873(119892119910 119892119909

119899 1199050))

(51)

thus

M119872119873

(119892119909 119892119910119899+1

120601 (1199050))

ge119871lowastT (M

119872119873(119892119909 119892119910

119899 1199050) lowastM

119872119873(119892119910 119892119909

119899 1199050))

(52)


M119872119873

(119892119909 119910 120601 (1199050))

ge119871lowastT (M

119872119873(119892119909 119910 119905

0) M119872119873

(119892119910 119909 1199050))

(53)


M119872119873

(119892119910 119909 120601 (1199050))

ge119871lowastT (M

119872119873(119892119909 119910 119905

0) M119872119873

(119892119910 119909 1199050))

(54)


T (M119872119873

(119892119909 119910 120601 (1199050)) M

119872119873(119892119910 119909 120601 (119905

0)))

ge119871lowastT ([M

119872119873(119892119909 119910 119905

0)]2

[M119872119873

(119892119910 119909 1199050)]2

) (55)


T (M119872119873

(119892119909 119910 120601119899

(1199050)) M

119872119873(119892119910 119909 120601

119899

(1199050)))

ge119871lowastT ([M

119872119873(119892119909 119910 120601

119899minus1

(1199050))]2

[M119872119873

(119892119910 119909 120601119899minus1

(1199050))]2

)

ge119871lowastT ([M

119872119873(119892119909 119910 119905

0)]2119899

[M119872119873

(119892119910 119909 1199050)]2119899

)

(56)


Thus

T (M119872119873

(119892119909 119910 120601119899

(1199050)) M

119872119873(119892119910 119909 120601

119899

(1199050)))

ge119871lowastT ([M

119872119873(119892119909 119910 119905

0)]2119899

[M119872119873

(119892119910 119909 1199050)]2119899

)

(57)


T (M119872119873

(119892119909 119910 119905) M119872119873

(119892119910 119909 119905))

ge119871lowastT(M

119872119873(119892119909 119910

infin

sum119896=1198990

120601119896

(1199050))

M119872119873

(119892119910 119909infin

sum119896=1198990

120601119896

(1199050)))

ge119871lowastT (M

119872119873(119892119909 119910 120601

1198990 (1199050)) M

119872119873(119892119910 119909 120601

1198990 (1199050)))

ge119871lowastT ([M

119872119873(119892119909 119910 119905

0)]21198990

[M119872119873

(119892119910 119909 1199050)]21198990

)

ge119871lowastT21198990+1

minus1

(119873119904(120583) 120583)

ge119871lowast (119873119904(120582) 120582)

(58)


T (M119872119873

(119892119909 119910 119905) M119872119873

(119892119910 119909 119905)) ge119871lowast (119873119904(120582) 120582)

forall119905 gt 0

(59)




T119898minus1

(119873119904(120583) 120583) ge



M119872119873



M119872119873

(119909 119910 1199050) ge119871lowast (119873119904(120583) 120583) (61)



0isin 119873

such that

119905 gtinfin

sum119896=1198990

120601119896

(1199050) (62)


M119872119873

(119892119909119899+1

119892119910119899+1

120601 (1199050))

= M119872119873

(119865 (119909119899 119910119899) 119865 (119910

119899 119909119899) 120601 (119905

0))

ge119871lowastT (M

119872119873(119892119909119899 119892119910119899 1199050) M119872119873

(119892119910119899 119892119909119899 1199050))

(63)

thus

M119872119873

(119892119909119899+1

119892119910119899+1

120601 (1199050))

ge119871lowastT (M

119872119873(119892119909119899 119892119910119899 1199050)

M119872119873

(119892119910119899 119892119909119899 1199050))

(64)


M119872119873

(119909 119910 120601 (1199050))

ge119871lowastT (M

119872119873(119909 119910 119905

0) M119872119873

(119910 119909 1199050))

(65)


M119872119873

(119909 119910 119905)

ge119871lowastM119872119873

(119909 119910infin

sum119896=1198990

120601119896

(1199050))

ge119871lowast M119872119873

(119909 119910 1206011198990 (1199050))

ge119871lowastT ([M

119872119873(119909 119910 119905

0)]21198990minus1

[M119872119873

(119910 119909 1199050)]21198990minus1

)

ge119871lowastT21198990+1

minus3

(119873119904(120583) 120583)

ge119871lowast (119873119904(120582) 120582)

(66)



Corollary 32 Let (119883M119872119873


M119872119873

(119865 (119909 119910) 119865 (119906 V) 120601 (119905))

ge119871lowastT (M

119872119873(119909 119906 119905) M

119872119873(119910 V 119905))

(67)



Corollary 33 Let (119883M119872119873




M119872119873

(119865 (119909 119910) 119865 (119906 V) 119896119905)

ge119871lowastT (M

119872119873(119892119909 119892119906 119905) M

119872119873(119892119910 119892V 119905))

(68)








1 1198871max119886

2 1198872) for all 119886 = (119886

1 1198862) and


M119872119873

(119909 119910 119905) = (119905

119905 +1003816100381610038161003816119909 minus 119910

1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119905 +1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)

forall119909 119910 isin 119883 119905 gt 0

(69)

Then (119883M119872119873


119892 (119909) =

0 119909 = 0

1 119909 =1

2119899 + 1

1

2119899 + 1 119909 =

1

2119899

119865 (119909 119910) =

1

(2119899 + 1)4 (119909 119910) = (

1

21198991

2119899)

0 otherwise

(70)

Let 119909119899= 119910119899= 12119899 We have

119892119909119899=

1

2119899 + 1997888rarr 0

119865 (119909119899 119910119899) =

1


(71)

but

M119872119873

(119865 (119892119909119899 119892119910119899) 119892119865 (119909

119899 119910119899) 119905)

= M119872119873

(0 1 119905) 999424999426999456 1119871lowast

(72)


119905

119883 + 119905ge min 119905

119884 + 119905

119905

119885 + 119905

119883

119883 + 119905le max 119884

119884 + 119905

119885

119885 + 119905


(73)



M119872119873

(119865 (119909 119910) 119865 (119906 V) 120601 (119905))

ge119871lowastT (M

119872119873(119892119909 119892119906 119905) M

119872119873(119892119910 119892V 119905))

(74)


21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)


1003816100381610038161003816 (75)





21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816 =2

(2119899 + 1)4

max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V

1003816100381610038161003816 =2119899

2119899 + 1

(76)




21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816 =2

(2119899 + 1)4

max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V

1003816100381610038161003816

= max 1

2119899 + 110038161003816100381610038161003816100381610038161003816

1

2119895 + 1minus

1

2119899 + 1

10038161003816100381610038161003816100381610038161003816

(77)

or

max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V

1003816100381610038161003816

= max 2119899

2119899 + 110038161003816100381610038161003816100381610038161003816

1

2119895 + 1minus

1

2119899 + 1

10038161003816100381610038161003816100381610038161003816

(78)

So (75) holds


21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816 =2

(2119899 + 1)4

max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V

1003816100381610038161003816

= max1003816100381610038161003816100381610038161003816

1

2119894 + 1minus

1

2119899 + 1

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1

2119895 + 1minus

1

2119899 + 1

10038161003816100381610038161003816100381610038161003816

(79)



21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816 = 2

100381610038161003816100381610038161003816100381610038161003816

1

(2119894 + 1)4minus

1

(2119899 + 1)4

100381610038161003816100381610038161003816100381610038161003816

max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V

1003816100381610038161003816

=1

2119894 + 1minus

1

2119899 + 1

(80)




119909 (119901) = int119887

119886

(1198701(119901 119902) + 119870

2(119901 119902))

times [119891 (119902 119909 (119902)) + 119892 (119902 119909 (119902))] 119889119902 + ℎ (119901)

(81)




(ii120579) 120579(119901) le 119901




1198701(119901 119902) ge 0 119870

2(119901 119902) le 0 forall119901 119902 isin 119868 (82)


0 le 119891 (119902 119909) minus 119891 (119902 119910) le 120582120579 (119909 minus 119910)


(iii) Consider

max 120582 120583 sup119901isin119868

int119887

119886

[1198701(119901 119902) minus 119870

2(119901 119902)] 119889119902 le

1

8 (84)






M119872119873

(119909 119910 119905) = (119905

119905 +1003816100381610038161003816119909 minus 119910

1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119905 +1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)

forall119909 119910 isin 119883 119905 gt 0

(85)


1 1198871max119886

2 1198872) for all 119886 = (119886

1 1198862) and 119887 = (119887

1 1198872) isin


119865 (119909 119910) (119901) = int119887

119886

1198701(119901 119902) [119891 (119902 119909 (119902)) + 119892 (119902 119910 (119902))] 119889119902

+ int119887

119886

1198702(119901 119902) [119891 (119902 119910 (119902))

+119892 (119902 119909 (119902))] 119889119902 + ℎ (119901)

(86)


119865 (119909 119910) (119901) minus 119865 (119906 V) (119901)

= int119887

119886

1198701(119901 119902) [119891 (119902 119909 (119902)) + 119892 (119902 119910 (119902))] 119889119902

+ int119887

119886

1198702(119901 119902) [119891 (119902 119910 (119902)) + 119892 (119902 119909 (119902))] 119889119902

minus int119887

119886

1198701(119901 119902) [119891 (119902 119906 (119902)) + 119892 (119902 V (119902))] 119889119902

minus int119887

119886

1198702(119901 119902) [119891 (119902 V (119902)) + 119892 (119902 119906 (119902))] 119889119902

= int119887

119886

1198701(119901 119902) [119891 (119902 119909 (119902)) minus 119891 (119902 119906 (119902))

+119892 (119902 119910 (119902)) minus 119892 (119902 V (119902))] 119889119902

+ int119887

119886

1198702(119901 119902) [119891 (119902 119910 (119902)) minus 119891 (119902 V (119902))

+119892 (119902 119909 (119902)) minus 119892 (119902 119906 (119902))] 119889119902

= int119887

119886

1198701(119901 119902) [(119891 (119902 119909 (119902)) minus 119891 (119902 119906 (119902)))

minus (119892 (119902 V (119902)) minus 119892 (119902 119910 (119902)))] 119889119902

minus int119887

119886

1198702(119901 119902) [(119891 (119902 V (119902)) minus 119891 (119902 119910 (119902)))

minus (119892 (119902 119909 (119902)) minus 119892 (119902 119906 (119902)))] 119889119902

le int119887

119886

1198701(119901 119902) [120582120579 (119909 (119902) minus 119906 (119902))

+120583120579 (V (119902) minus 119910 (119902))] 119889119902


minus int119887

119886

1198702(119901 119902) [120582120579 (V (119902) minus 119910 (119902))

+120583120579 (119909 (119902) minus 119906 (119902))] 119889119902

(87)

Thus

119865 (119909 119910) (119901) minus 119865 (119906 V) (119901)

le int119887

119886

1198701(119901 119902) [120582120579 (119909 (119902) minus 119906 (119902))

+120583120579 (V (119902) minus 119910 (119902))] 119889119902

minus int119887

119886

1198702(119901 119902) [120582120579 (V (119902) minus 119910 (119902))

+120583120579 (119909 (119902) minus 119906 (119902))] 119889119902

(88)


120579 (119909 (119902) minus 119906 (119902)) le 120579 (1003816100381610038161003816119909 (119902) minus 119906 (119902)

1003816100381610038161003816)

120579 (V (119902) minus 119910 (119902)) le 120579 (1003816100381610038161003816V (119902) minus 119910 (119902)

1003816100381610038161003816) (89)


1003816100381610038161003816119865 (119909 119910) (119901) minus 119865 (119906 V) (119901)1003816100381610038161003816

le int119887

119886

1198701(119901 119902) [120582120579 (

1003816100381610038161003816119909 (119902) minus 119906 (119902)1003816100381610038161003816)

+120583120579 (1003816100381610038161003816V (119902) minus 119910 (119902)

1003816100381610038161003816)] 119889119902

minus int119887

119886

1198702(119901 119902) [120582120579 (

1003816100381610038161003816V (119902) minus 119910 (119902)1003816100381610038161003816)

+120583120579 (1003816100381610038161003816119909 (119902) minus 119906 (119902)

1003816100381610038161003816)] 119889119902

le int119887

119886

1198701(119901 119902) [max 120582 120583 120579 (1003816100381610038161003816119909 (119902) minus 119906 (119902)

1003816100381610038161003816)

+max 120582 120583 120579 (1003816100381610038161003816V (119902) minus 119910 (119902)1003816100381610038161003816)] 119889119902

minus int119887

119886

1198702(119901 119902) [max 120582 120583 120579 (1003816100381610038161003816V (119902) minus 119910 (119902)

1003816100381610038161003816)

+max 120582 120583 120579 (1003816100381610038161003816119909 (119902) minus 119906 (119902)1003816100381610038161003816)] 119889119902

(90)


1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)1003816100381610038161003816

le max 120582 120583 int119887

119886

[1198701(119901 119902) minus 119870

2(119901 119902)] 119889119902

sdot [120579 (1003816100381610038161003816119909 (119902)minus119906 (119902)

1003816100381610038161003816)+120579 (1003816100381610038161003816V (119902) minus 119910 (119902)

1003816100381610038161003816)]

le max 120582 120583 sup119901isin119868

int119887

119886

[1198701(119901 119902) minus 119870

2(119901 119902)] 119889119902

sdot [120579 (1003816100381610038161003816119909 (119902)minus119906 (119902)

1003816100381610038161003816)+120579 (1003816100381610038161003816V (119902) minus 119910 (119902)

1003816100381610038161003816)]

le120579 (|119909 minus 119906|) + 120579 (

1003816100381610038161003816V minus 1199101003816100381610038161003816)

8

(91)

Thus

21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816 le120579 (|119909 minus 119906|) + 120579 (

1003816100381610038161003816V minus 1199101003816100381610038161003816)

4 (92)



120579 (1003816100381610038161003816119910 minus V1003816100381610038161003816) le 120579 (|119909 minus 119906| +

1003816100381610038161003816119910 minus V1003816100381610038161003816) (93)


120579 (|119909 minus 119906|) + 120579 (1003816100381610038161003816119910 minus V1003816100381610038161003816)

2le 120579 (|119909 minus 119906| +

1003816100381610038161003816119910 minus V1003816100381610038161003816)

le |119909 minus 119906| +1003816100381610038161003816119910 minus V1003816100381610038161003816


(94)

and so

120579 (|119909 minus 119906|) + 120579 (1003816100381610038161003816119910 minus V1003816100381610038161003816)



21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)



M119872119873

(119865 (119909 119910) 119865 (119906 V) 120601 (119905))

= M119872119873

(119865 (119909 119910) 119865 (119906 V) 119905

2)

= (1199052

1199052 +1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816

1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)1003816100381610038161003816

1199052 +1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816)

= (119905

119905 + 21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816

21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816119905 + 2

1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)1003816100381610038161003816)

ge119871lowast (

119905

119905 +max |119909 minus 119906| 1003816100381610038161003816119910 minus V1003816100381610038161003816


)



119905 + |119909 minus 119906|

119905

119905 +1003816100381610038161003816119910 minus V1003816100381610038161003816

max |119909 minus 119906|

119905 + |119909 minus 119906|

1003816100381610038161003816119910 minus V1003816100381610038161003816119905 +

1003816100381610038161003816119910 minus V1003816100381610038161003816)

ge119871lowastT (M

119872119873(119909 119906 119905) M

119872119873(119910 V 119905))

(97)




References




























Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in





M119872119873

(119865 (119909 119910) 119865 (119906 V) 120601 (119905))

ge119871lowastT (M

119872119873(119892119909 119892119906 119905) M

119872119873(119892119910 119892V 119905))

(30)




1198921199091= 119865 (119909

0 1199100) 119892119910

1= 119865 (119910

0 1199090) (31)


and 119910119899 in119883 such that

119892119909119899+1

= 119865 (119909119899 119910119899) 119892119910

119899+1= 119865 (119910

119899 119909119899) forall119899 ge 0 (32)





T119898minus1

(119873119904(120583) 120583) ge



M119872119873



M119872119873

(1198921199090 1198921199091 1199050) ge119871lowast (119873119904(120583) 120583)

M119872119873

(1198921199100 1198921199101 1199050) ge119871lowast (119873119904(120583) 120583)

(34)



0isin 119873

such that

119905 gtinfin

sum119896=1198990

120601119896

(1199050) (35)


M119872119873

(1198921199091 1198921199092 120601 (1199050))

= M119872119873

(119865 (1199090 1199100) 119865 (119909

1 1199101) 120601 (119905

0))

ge119871lowastT (M

119872119873(1198921199090 1198921199091 1199050) M119872119873

(1198921199100 1198921199101 1199050))

M119872119873

(1198921199101 1198921199102 120601 (1199050))

= M119872119873

(119865 (1199100 1199090) 119865 (119910

1 1199091) 120601 (119905

0))

ge119871lowastT (M

119872119873(1198921199100 1198921199101 1199050) M119872119873

(1198921199090 1198921199091 1199050))

(36)


M119872119873

(1198921199092 1198921199093 1206012

(1199050))

= M119872119873

(119865 (1199091 1199101) 119865 (119909

2 1199102) 1206012

(1199050))

ge119871lowastT (M

119872119873(1198921199091 1198921199092 120601 (1199050))

M119872119873

(1198921199101 1198921199102 120601 (1199050)))

ge119871lowastT ([M

119872119873(1198921199090 1198921199091 1199050)]2

[M119872119873

(1198921199100 1198921199101 1199050)]2

)

M119872119873

(1198921199102 1198921199103 1206012

(1199050))

= M119872119873

(119865 (1199101 1199091) 119865 (119910

2 1199092) 1206012

(1199050))

ge119871lowastT (M

119872119873(1198921199101 1198921199102 120601 (1199050))

M119872119873

(1198921199091 1198921199092 120601 (1199050)))

ge119871lowastT ([M

119872119873(1198921199100 1198921199101 1199050)]2

[M119872119873

(1198921199090 1198921199091 1199050)]2

)

(37)


M119872119873

(119892119909119899 119892119909119899+1

120601119899

(1199050))

ge119871lowastT ([M

119872119873(1198921199090 1198921199091 1199050)]2119899minus1

[M119872119873

(1198921199100 1198921199101 1199050)]2119899minus1

)

M119872119873

(119892119910119899 119892119910119899+1

120601119899

(1199050))

ge119871lowastT ([M

119872119873(1198921199100 1198921199101 1199050)]2119899minus1

[M119872119873

(1198921199090 1198921199091 1199050)]2119899minus1

)

(38)


M119872119873

(119892119909119899 119892119909119898 119905)

ge119871lowastM119872119873

(119892119909119899 119892119909119898infin

sum119896=1198990

120601119896

(1199050))

ge119871lowastM119872119873

(119892119909119899 119892119909119898119898minus1

sum119896=119899

120601119896

(1199050))


(M119872119873

(119892119909119899 119892119909119899+1

120601119899

(1199050))

M119872119873

(119892119909119899+1

119892119909119899+2

120601119899+1

(1199050))

M119872119873

(119892119909119898minus1

119892119909119898 120601119898minus1

(1199050)))


(T ([M119872119873

(1198921199090 1198921199091 1199050)]2119899minus1

[M119872119873

(1198921199100 1198921199101 1199050)]2119899minus1

)


T ([M119872119873

(1198921199090 1198921199091 1199050)]2119899

[M119872119873

(1198921199100 1198921199101 1199050)]2119899

)

T ([M119872119873

(1198921199090 1198921199091 1199050)]2119898minus2

[M119872119873

(1198921199100 1198921199101 1199050)]2119898minus2

))

ge119871lowastT ([M

119872119873(1198921199090 1198921199091 1199050)]2119898minus1

minus2119899minus1

[M119872119873

(1198921199100 1198921199101 1199050)]2119898minus1

minus2119899minus1

)


minus2119899

minus1

(119873119904(120583) 120583)

ge119871lowast (119873119904(120582) 120582)

(39)

which implies that

M119872119873

(119892119909119899 119892119909119898 119905) gt119871lowast (119873119904(120582) 120582) (40)


119899 is


Cauchy sequence



lim119899rarrinfin

119892119909119899= lim119899rarrinfin

119865 (119909119899 119910119899) = 119892119886 = 119909

lim119899rarrinfin

119892119910119899= lim119899rarrinfin

119865 (119910119899 119909119899) = 119892119887 = 119910

(41)

From (30) we get

M119872119873

(119865 (119909119899 119910119899) 119865 (119886 119887) 120601 (119905))

ge119871lowastT (M

119872119873(119892119909119899 119892119886 119905) M

119872119873(119892119910119899 119892119887 119905))

(42)

SinceM119872119873


M119872119873

(119892119886 119865 (119886 119887) 120601 (119905)) = 1119871lowast (43)

which implies that

119865 (119886 119887) = 119892119886 = 119909 (44)


119865 (119887 119886) = 119892119887 = 119910 (45)


119892119865 (119886 119887) = 119865 (119892119886 119892119887) 119892119865 (119887 119886) = 119865 (119892119887 119892119886) (46)

which implies that

119892119909 = 119865 (119909 119910) 119892119910 = 119865 (119910 119909) (47)



T119898minus1

(119873119904(120583) 120583) ge



M119872119873



M119872119873

(119892119909 119910 1199050) ge119871lowast (119873119904(120583) 120583)

M119872119873

(119892119910 119909 1199050) ge119871lowast (119873119904(120583) 120583)

(49)



0isin 119873

such that

119905 gtinfin

sum119896=1198990

120601119896

(1199050) (50)

Since

M119872119873

(119892119909 119892119910119899+1

120601 (1199050))

= M119872119873

(119865 (119909 119910) 119865 (119910119899 119909119899) 120601 (119905

0))

ge119871lowastT (M

119872119873(119892119909 119892119910

119899 1199050) lowastM

119872119873(119892119910 119892119909

119899 1199050))

(51)

thus

M119872119873

(119892119909 119892119910119899+1

120601 (1199050))

ge119871lowastT (M

119872119873(119892119909 119892119910

119899 1199050) lowastM

119872119873(119892119910 119892119909

119899 1199050))

(52)


M119872119873

(119892119909 119910 120601 (1199050))

ge119871lowastT (M

119872119873(119892119909 119910 119905

0) M119872119873

(119892119910 119909 1199050))

(53)


M119872119873

(119892119910 119909 120601 (1199050))

ge119871lowastT (M

119872119873(119892119909 119910 119905

0) M119872119873

(119892119910 119909 1199050))

(54)


T (M119872119873

(119892119909 119910 120601 (1199050)) M

119872119873(119892119910 119909 120601 (119905

0)))

ge119871lowastT ([M

119872119873(119892119909 119910 119905

0)]2

[M119872119873

(119892119910 119909 1199050)]2

) (55)


T (M119872119873

(119892119909 119910 120601119899

(1199050)) M

119872119873(119892119910 119909 120601

119899

(1199050)))

ge119871lowastT ([M

119872119873(119892119909 119910 120601

119899minus1

(1199050))]2

[M119872119873

(119892119910 119909 120601119899minus1

(1199050))]2

)

ge119871lowastT ([M

119872119873(119892119909 119910 119905

0)]2119899

[M119872119873

(119892119910 119909 1199050)]2119899

)

(56)


Thus

T (M119872119873

(119892119909 119910 120601119899

(1199050)) M

119872119873(119892119910 119909 120601

119899

(1199050)))

ge119871lowastT ([M

119872119873(119892119909 119910 119905

0)]2119899

[M119872119873

(119892119910 119909 1199050)]2119899

)

(57)


T (M119872119873

(119892119909 119910 119905) M119872119873

(119892119910 119909 119905))

ge119871lowastT(M

119872119873(119892119909 119910

infin

sum119896=1198990

120601119896

(1199050))

M119872119873

(119892119910 119909infin

sum119896=1198990

120601119896

(1199050)))

ge119871lowastT (M

119872119873(119892119909 119910 120601

1198990 (1199050)) M

119872119873(119892119910 119909 120601

1198990 (1199050)))

ge119871lowastT ([M

119872119873(119892119909 119910 119905

0)]21198990

[M119872119873

(119892119910 119909 1199050)]21198990

)

ge119871lowastT21198990+1

minus1

(119873119904(120583) 120583)

ge119871lowast (119873119904(120582) 120582)

(58)


T (M119872119873

(119892119909 119910 119905) M119872119873

(119892119910 119909 119905)) ge119871lowast (119873119904(120582) 120582)

forall119905 gt 0

(59)




T119898minus1

(119873119904(120583) 120583) ge



M119872119873



M119872119873

(119909 119910 1199050) ge119871lowast (119873119904(120583) 120583) (61)



0isin 119873

such that

119905 gtinfin

sum119896=1198990

120601119896

(1199050) (62)


M119872119873

(119892119909119899+1

119892119910119899+1

120601 (1199050))

= M119872119873

(119865 (119909119899 119910119899) 119865 (119910

119899 119909119899) 120601 (119905

0))

ge119871lowastT (M

119872119873(119892119909119899 119892119910119899 1199050) M119872119873

(119892119910119899 119892119909119899 1199050))

(63)

thus

M119872119873

(119892119909119899+1

119892119910119899+1

120601 (1199050))

ge119871lowastT (M

119872119873(119892119909119899 119892119910119899 1199050)

M119872119873

(119892119910119899 119892119909119899 1199050))

(64)


M119872119873

(119909 119910 120601 (1199050))

ge119871lowastT (M

119872119873(119909 119910 119905

0) M119872119873

(119910 119909 1199050))

(65)


M119872119873

(119909 119910 119905)

ge119871lowastM119872119873

(119909 119910infin

sum119896=1198990

120601119896

(1199050))

ge119871lowast M119872119873

(119909 119910 1206011198990 (1199050))

ge119871lowastT ([M

119872119873(119909 119910 119905

0)]21198990minus1

[M119872119873

(119910 119909 1199050)]21198990minus1

)

ge119871lowastT21198990+1

minus3

(119873119904(120583) 120583)

ge119871lowast (119873119904(120582) 120582)

(66)



Corollary 32 Let (119883M119872119873


M119872119873

(119865 (119909 119910) 119865 (119906 V) 120601 (119905))

ge119871lowastT (M

119872119873(119909 119906 119905) M

119872119873(119910 V 119905))

(67)



Corollary 33 Let (119883M119872119873




M119872119873

(119865 (119909 119910) 119865 (119906 V) 119896119905)

ge119871lowastT (M

119872119873(119892119909 119892119906 119905) M

119872119873(119892119910 119892V 119905))

(68)








1 1198871max119886

2 1198872) for all 119886 = (119886

1 1198862) and


M119872119873

(119909 119910 119905) = (119905

119905 +1003816100381610038161003816119909 minus 119910

1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119905 +1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)

forall119909 119910 isin 119883 119905 gt 0

(69)

Then (119883M119872119873


119892 (119909) =

0 119909 = 0

1 119909 =1

2119899 + 1

1

2119899 + 1 119909 =

1

2119899

119865 (119909 119910) =

1

(2119899 + 1)4 (119909 119910) = (

1

21198991

2119899)

0 otherwise

(70)

Let 119909119899= 119910119899= 12119899 We have

119892119909119899=

1

2119899 + 1997888rarr 0

119865 (119909119899 119910119899) =

1


(71)

but

M119872119873

(119865 (119892119909119899 119892119910119899) 119892119865 (119909

119899 119910119899) 119905)

= M119872119873

(0 1 119905) 999424999426999456 1119871lowast

(72)


119905

119883 + 119905ge min 119905

119884 + 119905

119905

119885 + 119905

119883

119883 + 119905le max 119884

119884 + 119905

119885

119885 + 119905


(73)



M119872119873

(119865 (119909 119910) 119865 (119906 V) 120601 (119905))

ge119871lowastT (M

119872119873(119892119909 119892119906 119905) M

119872119873(119892119910 119892V 119905))

(74)


21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)


1003816100381610038161003816 (75)





21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816 =2

(2119899 + 1)4

max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V

1003816100381610038161003816 =2119899

2119899 + 1

(76)




21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816 =2

(2119899 + 1)4

max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V

1003816100381610038161003816

= max 1

2119899 + 110038161003816100381610038161003816100381610038161003816

1

2119895 + 1minus

1

2119899 + 1

10038161003816100381610038161003816100381610038161003816

(77)

or

max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V

1003816100381610038161003816

= max 2119899

2119899 + 110038161003816100381610038161003816100381610038161003816

1

2119895 + 1minus

1

2119899 + 1

10038161003816100381610038161003816100381610038161003816

(78)

So (75) holds


21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816 =2

(2119899 + 1)4

max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V

1003816100381610038161003816

= max1003816100381610038161003816100381610038161003816

1

2119894 + 1minus

1

2119899 + 1

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1

2119895 + 1minus

1

2119899 + 1

10038161003816100381610038161003816100381610038161003816

(79)



21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816 = 2

100381610038161003816100381610038161003816100381610038161003816

1

(2119894 + 1)4minus

1

(2119899 + 1)4

100381610038161003816100381610038161003816100381610038161003816

max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V

1003816100381610038161003816

=1

2119894 + 1minus

1

2119899 + 1

(80)




119909 (119901) = int119887

119886

(1198701(119901 119902) + 119870

2(119901 119902))

times [119891 (119902 119909 (119902)) + 119892 (119902 119909 (119902))] 119889119902 + ℎ (119901)

(81)




(ii120579) 120579(119901) le 119901




1198701(119901 119902) ge 0 119870

2(119901 119902) le 0 forall119901 119902 isin 119868 (82)


0 le 119891 (119902 119909) minus 119891 (119902 119910) le 120582120579 (119909 minus 119910)


(iii) Consider

max 120582 120583 sup119901isin119868

int119887

119886

[1198701(119901 119902) minus 119870

2(119901 119902)] 119889119902 le

1

8 (84)






M119872119873

(119909 119910 119905) = (119905

119905 +1003816100381610038161003816119909 minus 119910

1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119905 +1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)

forall119909 119910 isin 119883 119905 gt 0

(85)


1 1198871max119886

2 1198872) for all 119886 = (119886

1 1198862) and 119887 = (119887

1 1198872) isin


119865 (119909 119910) (119901) = int119887

119886

1198701(119901 119902) [119891 (119902 119909 (119902)) + 119892 (119902 119910 (119902))] 119889119902

+ int119887

119886

1198702(119901 119902) [119891 (119902 119910 (119902))

+119892 (119902 119909 (119902))] 119889119902 + ℎ (119901)

(86)


119865 (119909 119910) (119901) minus 119865 (119906 V) (119901)

= int119887

119886

1198701(119901 119902) [119891 (119902 119909 (119902)) + 119892 (119902 119910 (119902))] 119889119902

+ int119887

119886

1198702(119901 119902) [119891 (119902 119910 (119902)) + 119892 (119902 119909 (119902))] 119889119902

minus int119887

119886

1198701(119901 119902) [119891 (119902 119906 (119902)) + 119892 (119902 V (119902))] 119889119902

minus int119887

119886

1198702(119901 119902) [119891 (119902 V (119902)) + 119892 (119902 119906 (119902))] 119889119902

= int119887

119886

1198701(119901 119902) [119891 (119902 119909 (119902)) minus 119891 (119902 119906 (119902))

+119892 (119902 119910 (119902)) minus 119892 (119902 V (119902))] 119889119902

+ int119887

119886

1198702(119901 119902) [119891 (119902 119910 (119902)) minus 119891 (119902 V (119902))

+119892 (119902 119909 (119902)) minus 119892 (119902 119906 (119902))] 119889119902

= int119887

119886

1198701(119901 119902) [(119891 (119902 119909 (119902)) minus 119891 (119902 119906 (119902)))

minus (119892 (119902 V (119902)) minus 119892 (119902 119910 (119902)))] 119889119902

minus int119887

119886

1198702(119901 119902) [(119891 (119902 V (119902)) minus 119891 (119902 119910 (119902)))

minus (119892 (119902 119909 (119902)) minus 119892 (119902 119906 (119902)))] 119889119902

le int119887

119886

1198701(119901 119902) [120582120579 (119909 (119902) minus 119906 (119902))

+120583120579 (V (119902) minus 119910 (119902))] 119889119902


minus int119887

119886

1198702(119901 119902) [120582120579 (V (119902) minus 119910 (119902))

+120583120579 (119909 (119902) minus 119906 (119902))] 119889119902

(87)

Thus

119865 (119909 119910) (119901) minus 119865 (119906 V) (119901)

le int119887

119886

1198701(119901 119902) [120582120579 (119909 (119902) minus 119906 (119902))

+120583120579 (V (119902) minus 119910 (119902))] 119889119902

minus int119887

119886

1198702(119901 119902) [120582120579 (V (119902) minus 119910 (119902))

+120583120579 (119909 (119902) minus 119906 (119902))] 119889119902

(88)


120579 (119909 (119902) minus 119906 (119902)) le 120579 (1003816100381610038161003816119909 (119902) minus 119906 (119902)

1003816100381610038161003816)

120579 (V (119902) minus 119910 (119902)) le 120579 (1003816100381610038161003816V (119902) minus 119910 (119902)

1003816100381610038161003816) (89)


1003816100381610038161003816119865 (119909 119910) (119901) minus 119865 (119906 V) (119901)1003816100381610038161003816

le int119887

119886

1198701(119901 119902) [120582120579 (

1003816100381610038161003816119909 (119902) minus 119906 (119902)1003816100381610038161003816)

+120583120579 (1003816100381610038161003816V (119902) minus 119910 (119902)

1003816100381610038161003816)] 119889119902

minus int119887

119886

1198702(119901 119902) [120582120579 (

1003816100381610038161003816V (119902) minus 119910 (119902)1003816100381610038161003816)

+120583120579 (1003816100381610038161003816119909 (119902) minus 119906 (119902)

1003816100381610038161003816)] 119889119902

le int119887

119886

1198701(119901 119902) [max 120582 120583 120579 (1003816100381610038161003816119909 (119902) minus 119906 (119902)

1003816100381610038161003816)

+max 120582 120583 120579 (1003816100381610038161003816V (119902) minus 119910 (119902)1003816100381610038161003816)] 119889119902

minus int119887

119886

1198702(119901 119902) [max 120582 120583 120579 (1003816100381610038161003816V (119902) minus 119910 (119902)

1003816100381610038161003816)

+max 120582 120583 120579 (1003816100381610038161003816119909 (119902) minus 119906 (119902)1003816100381610038161003816)] 119889119902

(90)


1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)1003816100381610038161003816

le max 120582 120583 int119887

119886

[1198701(119901 119902) minus 119870

2(119901 119902)] 119889119902

sdot [120579 (1003816100381610038161003816119909 (119902)minus119906 (119902)

1003816100381610038161003816)+120579 (1003816100381610038161003816V (119902) minus 119910 (119902)

1003816100381610038161003816)]

le max 120582 120583 sup119901isin119868

int119887

119886

[1198701(119901 119902) minus 119870

2(119901 119902)] 119889119902

sdot [120579 (1003816100381610038161003816119909 (119902)minus119906 (119902)

1003816100381610038161003816)+120579 (1003816100381610038161003816V (119902) minus 119910 (119902)

1003816100381610038161003816)]

le120579 (|119909 minus 119906|) + 120579 (

1003816100381610038161003816V minus 1199101003816100381610038161003816)

8

(91)

Thus

21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816 le120579 (|119909 minus 119906|) + 120579 (

1003816100381610038161003816V minus 1199101003816100381610038161003816)

4 (92)



120579 (1003816100381610038161003816119910 minus V1003816100381610038161003816) le 120579 (|119909 minus 119906| +

1003816100381610038161003816119910 minus V1003816100381610038161003816) (93)


120579 (|119909 minus 119906|) + 120579 (1003816100381610038161003816119910 minus V1003816100381610038161003816)

2le 120579 (|119909 minus 119906| +

1003816100381610038161003816119910 minus V1003816100381610038161003816)

le |119909 minus 119906| +1003816100381610038161003816119910 minus V1003816100381610038161003816


(94)

and so

120579 (|119909 minus 119906|) + 120579 (1003816100381610038161003816119910 minus V1003816100381610038161003816)



21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)



M119872119873

(119865 (119909 119910) 119865 (119906 V) 120601 (119905))

= M119872119873

(119865 (119909 119910) 119865 (119906 V) 119905

2)

= (1199052

1199052 +1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816

1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)1003816100381610038161003816

1199052 +1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816)

= (119905

119905 + 21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816

21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816119905 + 2

1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)1003816100381610038161003816)

ge119871lowast (

119905

119905 +max |119909 minus 119906| 1003816100381610038161003816119910 minus V1003816100381610038161003816


)



119905 + |119909 minus 119906|

119905

119905 +1003816100381610038161003816119910 minus V1003816100381610038161003816

max |119909 minus 119906|

119905 + |119909 minus 119906|

1003816100381610038161003816119910 minus V1003816100381610038161003816119905 +

1003816100381610038161003816119910 minus V1003816100381610038161003816)

ge119871lowastT (M

119872119873(119909 119906 119905) M

119872119873(119910 V 119905))

(97)




References




























Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




T ([M119872119873

(1198921199090 1198921199091 1199050)]2119899

[M119872119873

(1198921199100 1198921199101 1199050)]2119899

)

T ([M119872119873

(1198921199090 1198921199091 1199050)]2119898minus2

[M119872119873

(1198921199100 1198921199101 1199050)]2119898minus2

))

ge119871lowastT ([M

119872119873(1198921199090 1198921199091 1199050)]2119898minus1

minus2119899minus1

[M119872119873

(1198921199100 1198921199101 1199050)]2119898minus1

minus2119899minus1

)


minus2119899

minus1

(119873119904(120583) 120583)

ge119871lowast (119873119904(120582) 120582)

(39)

which implies that

M119872119873

(119892119909119899 119892119909119898 119905) gt119871lowast (119873119904(120582) 120582) (40)


119899 is


Cauchy sequence



lim119899rarrinfin

119892119909119899= lim119899rarrinfin

119865 (119909119899 119910119899) = 119892119886 = 119909

lim119899rarrinfin

119892119910119899= lim119899rarrinfin

119865 (119910119899 119909119899) = 119892119887 = 119910

(41)

From (30) we get

M119872119873

(119865 (119909119899 119910119899) 119865 (119886 119887) 120601 (119905))

ge119871lowastT (M

119872119873(119892119909119899 119892119886 119905) M

119872119873(119892119910119899 119892119887 119905))

(42)

SinceM119872119873


M119872119873

(119892119886 119865 (119886 119887) 120601 (119905)) = 1119871lowast (43)

which implies that

119865 (119886 119887) = 119892119886 = 119909 (44)


119865 (119887 119886) = 119892119887 = 119910 (45)


119892119865 (119886 119887) = 119865 (119892119886 119892119887) 119892119865 (119887 119886) = 119865 (119892119887 119892119886) (46)

which implies that

119892119909 = 119865 (119909 119910) 119892119910 = 119865 (119910 119909) (47)



T119898minus1

(119873119904(120583) 120583) ge



M119872119873



M119872119873

(119892119909 119910 1199050) ge119871lowast (119873119904(120583) 120583)

M119872119873

(119892119910 119909 1199050) ge119871lowast (119873119904(120583) 120583)

(49)



0isin 119873

such that

119905 gtinfin

sum119896=1198990

120601119896

(1199050) (50)

Since

M119872119873

(119892119909 119892119910119899+1

120601 (1199050))

= M119872119873

(119865 (119909 119910) 119865 (119910119899 119909119899) 120601 (119905

0))

ge119871lowastT (M

119872119873(119892119909 119892119910

119899 1199050) lowastM

119872119873(119892119910 119892119909

119899 1199050))

(51)

thus

M119872119873

(119892119909 119892119910119899+1

120601 (1199050))

ge119871lowastT (M

119872119873(119892119909 119892119910

119899 1199050) lowastM

119872119873(119892119910 119892119909

119899 1199050))

(52)


M119872119873

(119892119909 119910 120601 (1199050))

ge119871lowastT (M

119872119873(119892119909 119910 119905

0) M119872119873

(119892119910 119909 1199050))

(53)


M119872119873

(119892119910 119909 120601 (1199050))

ge119871lowastT (M

119872119873(119892119909 119910 119905

0) M119872119873

(119892119910 119909 1199050))

(54)


T (M119872119873

(119892119909 119910 120601 (1199050)) M

119872119873(119892119910 119909 120601 (119905

0)))

ge119871lowastT ([M

119872119873(119892119909 119910 119905

0)]2

[M119872119873

(119892119910 119909 1199050)]2

) (55)


T (M119872119873

(119892119909 119910 120601119899

(1199050)) M

119872119873(119892119910 119909 120601

119899

(1199050)))

ge119871lowastT ([M

119872119873(119892119909 119910 120601

119899minus1

(1199050))]2

[M119872119873

(119892119910 119909 120601119899minus1

(1199050))]2

)

ge119871lowastT ([M

119872119873(119892119909 119910 119905

0)]2119899

[M119872119873

(119892119910 119909 1199050)]2119899

)

(56)


Thus

T (M119872119873

(119892119909 119910 120601119899

(1199050)) M

119872119873(119892119910 119909 120601

119899

(1199050)))

ge119871lowastT ([M

119872119873(119892119909 119910 119905

0)]2119899

[M119872119873

(119892119910 119909 1199050)]2119899

)

(57)


T (M119872119873

(119892119909 119910 119905) M119872119873

(119892119910 119909 119905))

ge119871lowastT(M

119872119873(119892119909 119910

infin

sum119896=1198990

120601119896

(1199050))

M119872119873

(119892119910 119909infin

sum119896=1198990

120601119896

(1199050)))

ge119871lowastT (M

119872119873(119892119909 119910 120601

1198990 (1199050)) M

119872119873(119892119910 119909 120601

1198990 (1199050)))

ge119871lowastT ([M

119872119873(119892119909 119910 119905

0)]21198990

[M119872119873

(119892119910 119909 1199050)]21198990

)

ge119871lowastT21198990+1

minus1

(119873119904(120583) 120583)

ge119871lowast (119873119904(120582) 120582)

(58)


T (M119872119873

(119892119909 119910 119905) M119872119873

(119892119910 119909 119905)) ge119871lowast (119873119904(120582) 120582)

forall119905 gt 0

(59)




T119898minus1

(119873119904(120583) 120583) ge



M119872119873



M119872119873

(119909 119910 1199050) ge119871lowast (119873119904(120583) 120583) (61)



0isin 119873

such that

119905 gtinfin

sum119896=1198990

120601119896

(1199050) (62)


M119872119873

(119892119909119899+1

119892119910119899+1

120601 (1199050))

= M119872119873

(119865 (119909119899 119910119899) 119865 (119910

119899 119909119899) 120601 (119905

0))

ge119871lowastT (M

119872119873(119892119909119899 119892119910119899 1199050) M119872119873

(119892119910119899 119892119909119899 1199050))

(63)

thus

M119872119873

(119892119909119899+1

119892119910119899+1

120601 (1199050))

ge119871lowastT (M

119872119873(119892119909119899 119892119910119899 1199050)

M119872119873

(119892119910119899 119892119909119899 1199050))

(64)


M119872119873

(119909 119910 120601 (1199050))

ge119871lowastT (M

119872119873(119909 119910 119905

0) M119872119873

(119910 119909 1199050))

(65)


M119872119873

(119909 119910 119905)

ge119871lowastM119872119873

(119909 119910infin

sum119896=1198990

120601119896

(1199050))

ge119871lowast M119872119873

(119909 119910 1206011198990 (1199050))

ge119871lowastT ([M

119872119873(119909 119910 119905

0)]21198990minus1

[M119872119873

(119910 119909 1199050)]21198990minus1

)

ge119871lowastT21198990+1

minus3

(119873119904(120583) 120583)

ge119871lowast (119873119904(120582) 120582)

(66)



Corollary 32 Let (119883M119872119873


M119872119873

(119865 (119909 119910) 119865 (119906 V) 120601 (119905))

ge119871lowastT (M

119872119873(119909 119906 119905) M

119872119873(119910 V 119905))

(67)



Corollary 33 Let (119883M119872119873




M119872119873

(119865 (119909 119910) 119865 (119906 V) 119896119905)

ge119871lowastT (M

119872119873(119892119909 119892119906 119905) M

119872119873(119892119910 119892V 119905))

(68)








1 1198871max119886

2 1198872) for all 119886 = (119886

1 1198862) and


M119872119873

(119909 119910 119905) = (119905

119905 +1003816100381610038161003816119909 minus 119910

1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119905 +1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)

forall119909 119910 isin 119883 119905 gt 0

(69)

Then (119883M119872119873


119892 (119909) =

0 119909 = 0

1 119909 =1

2119899 + 1

1

2119899 + 1 119909 =

1

2119899

119865 (119909 119910) =

1

(2119899 + 1)4 (119909 119910) = (

1

21198991

2119899)

0 otherwise

(70)

Let 119909119899= 119910119899= 12119899 We have

119892119909119899=

1

2119899 + 1997888rarr 0

119865 (119909119899 119910119899) =

1


(71)

but

M119872119873

(119865 (119892119909119899 119892119910119899) 119892119865 (119909

119899 119910119899) 119905)

= M119872119873

(0 1 119905) 999424999426999456 1119871lowast

(72)


119905

119883 + 119905ge min 119905

119884 + 119905

119905

119885 + 119905

119883

119883 + 119905le max 119884

119884 + 119905

119885

119885 + 119905


(73)



M119872119873

(119865 (119909 119910) 119865 (119906 V) 120601 (119905))

ge119871lowastT (M

119872119873(119892119909 119892119906 119905) M

119872119873(119892119910 119892V 119905))

(74)


21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)


1003816100381610038161003816 (75)





21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816 =2

(2119899 + 1)4

max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V

1003816100381610038161003816 =2119899

2119899 + 1

(76)




21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816 =2

(2119899 + 1)4

max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V

1003816100381610038161003816

= max 1

2119899 + 110038161003816100381610038161003816100381610038161003816

1

2119895 + 1minus

1

2119899 + 1

10038161003816100381610038161003816100381610038161003816

(77)

or

max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V

1003816100381610038161003816

= max 2119899

2119899 + 110038161003816100381610038161003816100381610038161003816

1

2119895 + 1minus

1

2119899 + 1

10038161003816100381610038161003816100381610038161003816

(78)

So (75) holds


21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816 =2

(2119899 + 1)4

max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V

1003816100381610038161003816

= max1003816100381610038161003816100381610038161003816

1

2119894 + 1minus

1

2119899 + 1

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1

2119895 + 1minus

1

2119899 + 1

10038161003816100381610038161003816100381610038161003816

(79)



21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816 = 2

100381610038161003816100381610038161003816100381610038161003816

1

(2119894 + 1)4minus

1

(2119899 + 1)4

100381610038161003816100381610038161003816100381610038161003816

max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V

1003816100381610038161003816

=1

2119894 + 1minus

1

2119899 + 1

(80)




119909 (119901) = int119887

119886

(1198701(119901 119902) + 119870

2(119901 119902))

times [119891 (119902 119909 (119902)) + 119892 (119902 119909 (119902))] 119889119902 + ℎ (119901)

(81)




(ii120579) 120579(119901) le 119901




1198701(119901 119902) ge 0 119870

2(119901 119902) le 0 forall119901 119902 isin 119868 (82)


0 le 119891 (119902 119909) minus 119891 (119902 119910) le 120582120579 (119909 minus 119910)


(iii) Consider

max 120582 120583 sup119901isin119868

int119887

119886

[1198701(119901 119902) minus 119870

2(119901 119902)] 119889119902 le

1

8 (84)






M119872119873

(119909 119910 119905) = (119905

119905 +1003816100381610038161003816119909 minus 119910

1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119905 +1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)

forall119909 119910 isin 119883 119905 gt 0

(85)


1 1198871max119886

2 1198872) for all 119886 = (119886

1 1198862) and 119887 = (119887

1 1198872) isin


119865 (119909 119910) (119901) = int119887

119886

1198701(119901 119902) [119891 (119902 119909 (119902)) + 119892 (119902 119910 (119902))] 119889119902

+ int119887

119886

1198702(119901 119902) [119891 (119902 119910 (119902))

+119892 (119902 119909 (119902))] 119889119902 + ℎ (119901)

(86)


119865 (119909 119910) (119901) minus 119865 (119906 V) (119901)

= int119887

119886

1198701(119901 119902) [119891 (119902 119909 (119902)) + 119892 (119902 119910 (119902))] 119889119902

+ int119887

119886

1198702(119901 119902) [119891 (119902 119910 (119902)) + 119892 (119902 119909 (119902))] 119889119902

minus int119887

119886

1198701(119901 119902) [119891 (119902 119906 (119902)) + 119892 (119902 V (119902))] 119889119902

minus int119887

119886

1198702(119901 119902) [119891 (119902 V (119902)) + 119892 (119902 119906 (119902))] 119889119902

= int119887

119886

1198701(119901 119902) [119891 (119902 119909 (119902)) minus 119891 (119902 119906 (119902))

+119892 (119902 119910 (119902)) minus 119892 (119902 V (119902))] 119889119902

+ int119887

119886

1198702(119901 119902) [119891 (119902 119910 (119902)) minus 119891 (119902 V (119902))

+119892 (119902 119909 (119902)) minus 119892 (119902 119906 (119902))] 119889119902

= int119887

119886

1198701(119901 119902) [(119891 (119902 119909 (119902)) minus 119891 (119902 119906 (119902)))

minus (119892 (119902 V (119902)) minus 119892 (119902 119910 (119902)))] 119889119902

minus int119887

119886

1198702(119901 119902) [(119891 (119902 V (119902)) minus 119891 (119902 119910 (119902)))

minus (119892 (119902 119909 (119902)) minus 119892 (119902 119906 (119902)))] 119889119902

le int119887

119886

1198701(119901 119902) [120582120579 (119909 (119902) minus 119906 (119902))

+120583120579 (V (119902) minus 119910 (119902))] 119889119902


minus int119887

119886

1198702(119901 119902) [120582120579 (V (119902) minus 119910 (119902))

+120583120579 (119909 (119902) minus 119906 (119902))] 119889119902

(87)

Thus

119865 (119909 119910) (119901) minus 119865 (119906 V) (119901)

le int119887

119886

1198701(119901 119902) [120582120579 (119909 (119902) minus 119906 (119902))

+120583120579 (V (119902) minus 119910 (119902))] 119889119902

minus int119887

119886

1198702(119901 119902) [120582120579 (V (119902) minus 119910 (119902))

+120583120579 (119909 (119902) minus 119906 (119902))] 119889119902

(88)


120579 (119909 (119902) minus 119906 (119902)) le 120579 (1003816100381610038161003816119909 (119902) minus 119906 (119902)

1003816100381610038161003816)

120579 (V (119902) minus 119910 (119902)) le 120579 (1003816100381610038161003816V (119902) minus 119910 (119902)

1003816100381610038161003816) (89)


1003816100381610038161003816119865 (119909 119910) (119901) minus 119865 (119906 V) (119901)1003816100381610038161003816

le int119887

119886

1198701(119901 119902) [120582120579 (

1003816100381610038161003816119909 (119902) minus 119906 (119902)1003816100381610038161003816)

+120583120579 (1003816100381610038161003816V (119902) minus 119910 (119902)

1003816100381610038161003816)] 119889119902

minus int119887

119886

1198702(119901 119902) [120582120579 (

1003816100381610038161003816V (119902) minus 119910 (119902)1003816100381610038161003816)

+120583120579 (1003816100381610038161003816119909 (119902) minus 119906 (119902)

1003816100381610038161003816)] 119889119902

le int119887

119886

1198701(119901 119902) [max 120582 120583 120579 (1003816100381610038161003816119909 (119902) minus 119906 (119902)

1003816100381610038161003816)

+max 120582 120583 120579 (1003816100381610038161003816V (119902) minus 119910 (119902)1003816100381610038161003816)] 119889119902

minus int119887

119886

1198702(119901 119902) [max 120582 120583 120579 (1003816100381610038161003816V (119902) minus 119910 (119902)

1003816100381610038161003816)

+max 120582 120583 120579 (1003816100381610038161003816119909 (119902) minus 119906 (119902)1003816100381610038161003816)] 119889119902

(90)


1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)1003816100381610038161003816

le max 120582 120583 int119887

119886

[1198701(119901 119902) minus 119870

2(119901 119902)] 119889119902

sdot [120579 (1003816100381610038161003816119909 (119902)minus119906 (119902)

1003816100381610038161003816)+120579 (1003816100381610038161003816V (119902) minus 119910 (119902)

1003816100381610038161003816)]

le max 120582 120583 sup119901isin119868

int119887

119886

[1198701(119901 119902) minus 119870

2(119901 119902)] 119889119902

sdot [120579 (1003816100381610038161003816119909 (119902)minus119906 (119902)

1003816100381610038161003816)+120579 (1003816100381610038161003816V (119902) minus 119910 (119902)

1003816100381610038161003816)]

le120579 (|119909 minus 119906|) + 120579 (

1003816100381610038161003816V minus 1199101003816100381610038161003816)

8

(91)

Thus

21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816 le120579 (|119909 minus 119906|) + 120579 (

1003816100381610038161003816V minus 1199101003816100381610038161003816)

4 (92)



120579 (1003816100381610038161003816119910 minus V1003816100381610038161003816) le 120579 (|119909 minus 119906| +

1003816100381610038161003816119910 minus V1003816100381610038161003816) (93)


120579 (|119909 minus 119906|) + 120579 (1003816100381610038161003816119910 minus V1003816100381610038161003816)

2le 120579 (|119909 minus 119906| +

1003816100381610038161003816119910 minus V1003816100381610038161003816)

le |119909 minus 119906| +1003816100381610038161003816119910 minus V1003816100381610038161003816


(94)

and so

120579 (|119909 minus 119906|) + 120579 (1003816100381610038161003816119910 minus V1003816100381610038161003816)



21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)



M119872119873

(119865 (119909 119910) 119865 (119906 V) 120601 (119905))

= M119872119873

(119865 (119909 119910) 119865 (119906 V) 119905

2)

= (1199052

1199052 +1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816

1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)1003816100381610038161003816

1199052 +1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816)

= (119905

119905 + 21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816

21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816119905 + 2

1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)1003816100381610038161003816)

ge119871lowast (

119905

119905 +max |119909 minus 119906| 1003816100381610038161003816119910 minus V1003816100381610038161003816


)



119905 + |119909 minus 119906|

119905

119905 +1003816100381610038161003816119910 minus V1003816100381610038161003816

max |119909 minus 119906|

119905 + |119909 minus 119906|

1003816100381610038161003816119910 minus V1003816100381610038161003816119905 +

1003816100381610038161003816119910 minus V1003816100381610038161003816)

ge119871lowastT (M

119872119873(119909 119906 119905) M

119872119873(119910 V 119905))

(97)




References




























Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




Thus

T (M119872119873

(119892119909 119910 120601119899

(1199050)) M

119872119873(119892119910 119909 120601

119899

(1199050)))

ge119871lowastT ([M

119872119873(119892119909 119910 119905

0)]2119899

[M119872119873

(119892119910 119909 1199050)]2119899

)

(57)


T (M119872119873

(119892119909 119910 119905) M119872119873

(119892119910 119909 119905))

ge119871lowastT(M

119872119873(119892119909 119910

infin

sum119896=1198990

120601119896

(1199050))

M119872119873

(119892119910 119909infin

sum119896=1198990

120601119896

(1199050)))

ge119871lowastT (M

119872119873(119892119909 119910 120601

1198990 (1199050)) M

119872119873(119892119910 119909 120601

1198990 (1199050)))

ge119871lowastT ([M

119872119873(119892119909 119910 119905

0)]21198990

[M119872119873

(119892119910 119909 1199050)]21198990

)

ge119871lowastT21198990+1

minus1

(119873119904(120583) 120583)

ge119871lowast (119873119904(120582) 120582)

(58)


T (M119872119873

(119892119909 119910 119905) M119872119873

(119892119910 119909 119905)) ge119871lowast (119873119904(120582) 120582)

forall119905 gt 0

(59)




T119898minus1

(119873119904(120583) 120583) ge



M119872119873



M119872119873

(119909 119910 1199050) ge119871lowast (119873119904(120583) 120583) (61)



0isin 119873

such that

119905 gtinfin

sum119896=1198990

120601119896

(1199050) (62)


M119872119873

(119892119909119899+1

119892119910119899+1

120601 (1199050))

= M119872119873

(119865 (119909119899 119910119899) 119865 (119910

119899 119909119899) 120601 (119905

0))

ge119871lowastT (M

119872119873(119892119909119899 119892119910119899 1199050) M119872119873

(119892119910119899 119892119909119899 1199050))

(63)

thus

M119872119873

(119892119909119899+1

119892119910119899+1

120601 (1199050))

ge119871lowastT (M

119872119873(119892119909119899 119892119910119899 1199050)

M119872119873

(119892119910119899 119892119909119899 1199050))

(64)


M119872119873

(119909 119910 120601 (1199050))

ge119871lowastT (M

119872119873(119909 119910 119905

0) M119872119873

(119910 119909 1199050))

(65)


M119872119873

(119909 119910 119905)

ge119871lowastM119872119873

(119909 119910infin

sum119896=1198990

120601119896

(1199050))

ge119871lowast M119872119873

(119909 119910 1206011198990 (1199050))

ge119871lowastT ([M

119872119873(119909 119910 119905

0)]21198990minus1

[M119872119873

(119910 119909 1199050)]21198990minus1

)

ge119871lowastT21198990+1

minus3

(119873119904(120583) 120583)

ge119871lowast (119873119904(120582) 120582)

(66)



Corollary 32 Let (119883M119872119873


M119872119873

(119865 (119909 119910) 119865 (119906 V) 120601 (119905))

ge119871lowastT (M

119872119873(119909 119906 119905) M

119872119873(119910 V 119905))

(67)



Corollary 33 Let (119883M119872119873




M119872119873

(119865 (119909 119910) 119865 (119906 V) 119896119905)

ge119871lowastT (M

119872119873(119892119909 119892119906 119905) M

119872119873(119892119910 119892V 119905))

(68)








1 1198871max119886

2 1198872) for all 119886 = (119886

1 1198862) and


M119872119873

(119909 119910 119905) = (119905

119905 +1003816100381610038161003816119909 minus 119910

1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119905 +1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)

forall119909 119910 isin 119883 119905 gt 0

(69)

Then (119883M119872119873


119892 (119909) =

0 119909 = 0

1 119909 =1

2119899 + 1

1

2119899 + 1 119909 =

1

2119899

119865 (119909 119910) =

1

(2119899 + 1)4 (119909 119910) = (

1

21198991

2119899)

0 otherwise

(70)

Let 119909119899= 119910119899= 12119899 We have

119892119909119899=

1

2119899 + 1997888rarr 0

119865 (119909119899 119910119899) =

1


(71)

but

M119872119873

(119865 (119892119909119899 119892119910119899) 119892119865 (119909

119899 119910119899) 119905)

= M119872119873

(0 1 119905) 999424999426999456 1119871lowast

(72)


119905

119883 + 119905ge min 119905

119884 + 119905

119905

119885 + 119905

119883

119883 + 119905le max 119884

119884 + 119905

119885

119885 + 119905


(73)



M119872119873

(119865 (119909 119910) 119865 (119906 V) 120601 (119905))

ge119871lowastT (M

119872119873(119892119909 119892119906 119905) M

119872119873(119892119910 119892V 119905))

(74)


21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)


1003816100381610038161003816 (75)





21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816 =2

(2119899 + 1)4

max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V

1003816100381610038161003816 =2119899

2119899 + 1

(76)




21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816 =2

(2119899 + 1)4

max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V

1003816100381610038161003816

= max 1

2119899 + 110038161003816100381610038161003816100381610038161003816

1

2119895 + 1minus

1

2119899 + 1

10038161003816100381610038161003816100381610038161003816

(77)

or

max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V

1003816100381610038161003816

= max 2119899

2119899 + 110038161003816100381610038161003816100381610038161003816

1

2119895 + 1minus

1

2119899 + 1

10038161003816100381610038161003816100381610038161003816

(78)

So (75) holds


21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816 =2

(2119899 + 1)4

max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V

1003816100381610038161003816

= max1003816100381610038161003816100381610038161003816

1

2119894 + 1minus

1

2119899 + 1

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1

2119895 + 1minus

1

2119899 + 1

10038161003816100381610038161003816100381610038161003816

(79)



21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816 = 2

100381610038161003816100381610038161003816100381610038161003816

1

(2119894 + 1)4minus

1

(2119899 + 1)4

100381610038161003816100381610038161003816100381610038161003816

max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V

1003816100381610038161003816

=1

2119894 + 1minus

1

2119899 + 1

(80)




119909 (119901) = int119887

119886

(1198701(119901 119902) + 119870

2(119901 119902))

times [119891 (119902 119909 (119902)) + 119892 (119902 119909 (119902))] 119889119902 + ℎ (119901)

(81)




(ii120579) 120579(119901) le 119901




1198701(119901 119902) ge 0 119870

2(119901 119902) le 0 forall119901 119902 isin 119868 (82)


0 le 119891 (119902 119909) minus 119891 (119902 119910) le 120582120579 (119909 minus 119910)


(iii) Consider

max 120582 120583 sup119901isin119868

int119887

119886

[1198701(119901 119902) minus 119870

2(119901 119902)] 119889119902 le

1

8 (84)






M119872119873

(119909 119910 119905) = (119905

119905 +1003816100381610038161003816119909 minus 119910

1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119905 +1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)

forall119909 119910 isin 119883 119905 gt 0

(85)


1 1198871max119886

2 1198872) for all 119886 = (119886

1 1198862) and 119887 = (119887

1 1198872) isin


119865 (119909 119910) (119901) = int119887

119886

1198701(119901 119902) [119891 (119902 119909 (119902)) + 119892 (119902 119910 (119902))] 119889119902

+ int119887

119886

1198702(119901 119902) [119891 (119902 119910 (119902))

+119892 (119902 119909 (119902))] 119889119902 + ℎ (119901)

(86)


119865 (119909 119910) (119901) minus 119865 (119906 V) (119901)

= int119887

119886

1198701(119901 119902) [119891 (119902 119909 (119902)) + 119892 (119902 119910 (119902))] 119889119902

+ int119887

119886

1198702(119901 119902) [119891 (119902 119910 (119902)) + 119892 (119902 119909 (119902))] 119889119902

minus int119887

119886

1198701(119901 119902) [119891 (119902 119906 (119902)) + 119892 (119902 V (119902))] 119889119902

minus int119887

119886

1198702(119901 119902) [119891 (119902 V (119902)) + 119892 (119902 119906 (119902))] 119889119902

= int119887

119886

1198701(119901 119902) [119891 (119902 119909 (119902)) minus 119891 (119902 119906 (119902))

+119892 (119902 119910 (119902)) minus 119892 (119902 V (119902))] 119889119902

+ int119887

119886

1198702(119901 119902) [119891 (119902 119910 (119902)) minus 119891 (119902 V (119902))

+119892 (119902 119909 (119902)) minus 119892 (119902 119906 (119902))] 119889119902

= int119887

119886

1198701(119901 119902) [(119891 (119902 119909 (119902)) minus 119891 (119902 119906 (119902)))

minus (119892 (119902 V (119902)) minus 119892 (119902 119910 (119902)))] 119889119902

minus int119887

119886

1198702(119901 119902) [(119891 (119902 V (119902)) minus 119891 (119902 119910 (119902)))

minus (119892 (119902 119909 (119902)) minus 119892 (119902 119906 (119902)))] 119889119902

le int119887

119886

1198701(119901 119902) [120582120579 (119909 (119902) minus 119906 (119902))

+120583120579 (V (119902) minus 119910 (119902))] 119889119902


minus int119887

119886

1198702(119901 119902) [120582120579 (V (119902) minus 119910 (119902))

+120583120579 (119909 (119902) minus 119906 (119902))] 119889119902

(87)

Thus

119865 (119909 119910) (119901) minus 119865 (119906 V) (119901)

le int119887

119886

1198701(119901 119902) [120582120579 (119909 (119902) minus 119906 (119902))

+120583120579 (V (119902) minus 119910 (119902))] 119889119902

minus int119887

119886

1198702(119901 119902) [120582120579 (V (119902) minus 119910 (119902))

+120583120579 (119909 (119902) minus 119906 (119902))] 119889119902

(88)


120579 (119909 (119902) minus 119906 (119902)) le 120579 (1003816100381610038161003816119909 (119902) minus 119906 (119902)

1003816100381610038161003816)

120579 (V (119902) minus 119910 (119902)) le 120579 (1003816100381610038161003816V (119902) minus 119910 (119902)

1003816100381610038161003816) (89)


1003816100381610038161003816119865 (119909 119910) (119901) minus 119865 (119906 V) (119901)1003816100381610038161003816

le int119887

119886

1198701(119901 119902) [120582120579 (

1003816100381610038161003816119909 (119902) minus 119906 (119902)1003816100381610038161003816)

+120583120579 (1003816100381610038161003816V (119902) minus 119910 (119902)

1003816100381610038161003816)] 119889119902

minus int119887

119886

1198702(119901 119902) [120582120579 (

1003816100381610038161003816V (119902) minus 119910 (119902)1003816100381610038161003816)

+120583120579 (1003816100381610038161003816119909 (119902) minus 119906 (119902)

1003816100381610038161003816)] 119889119902

le int119887

119886

1198701(119901 119902) [max 120582 120583 120579 (1003816100381610038161003816119909 (119902) minus 119906 (119902)

1003816100381610038161003816)

+max 120582 120583 120579 (1003816100381610038161003816V (119902) minus 119910 (119902)1003816100381610038161003816)] 119889119902

minus int119887

119886

1198702(119901 119902) [max 120582 120583 120579 (1003816100381610038161003816V (119902) minus 119910 (119902)

1003816100381610038161003816)

+max 120582 120583 120579 (1003816100381610038161003816119909 (119902) minus 119906 (119902)1003816100381610038161003816)] 119889119902

(90)


1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)1003816100381610038161003816

le max 120582 120583 int119887

119886

[1198701(119901 119902) minus 119870

2(119901 119902)] 119889119902

sdot [120579 (1003816100381610038161003816119909 (119902)minus119906 (119902)

1003816100381610038161003816)+120579 (1003816100381610038161003816V (119902) minus 119910 (119902)

1003816100381610038161003816)]

le max 120582 120583 sup119901isin119868

int119887

119886

[1198701(119901 119902) minus 119870

2(119901 119902)] 119889119902

sdot [120579 (1003816100381610038161003816119909 (119902)minus119906 (119902)

1003816100381610038161003816)+120579 (1003816100381610038161003816V (119902) minus 119910 (119902)

1003816100381610038161003816)]

le120579 (|119909 minus 119906|) + 120579 (

1003816100381610038161003816V minus 1199101003816100381610038161003816)

8

(91)

Thus

21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816 le120579 (|119909 minus 119906|) + 120579 (

1003816100381610038161003816V minus 1199101003816100381610038161003816)

4 (92)



120579 (1003816100381610038161003816119910 minus V1003816100381610038161003816) le 120579 (|119909 minus 119906| +

1003816100381610038161003816119910 minus V1003816100381610038161003816) (93)


120579 (|119909 minus 119906|) + 120579 (1003816100381610038161003816119910 minus V1003816100381610038161003816)

2le 120579 (|119909 minus 119906| +

1003816100381610038161003816119910 minus V1003816100381610038161003816)

le |119909 minus 119906| +1003816100381610038161003816119910 minus V1003816100381610038161003816


(94)

and so

120579 (|119909 minus 119906|) + 120579 (1003816100381610038161003816119910 minus V1003816100381610038161003816)



21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)



M119872119873

(119865 (119909 119910) 119865 (119906 V) 120601 (119905))

= M119872119873

(119865 (119909 119910) 119865 (119906 V) 119905

2)

= (1199052

1199052 +1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816

1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)1003816100381610038161003816

1199052 +1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816)

= (119905

119905 + 21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816

21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816119905 + 2

1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)1003816100381610038161003816)

ge119871lowast (

119905

119905 +max |119909 minus 119906| 1003816100381610038161003816119910 minus V1003816100381610038161003816


)



119905 + |119909 minus 119906|

119905

119905 +1003816100381610038161003816119910 minus V1003816100381610038161003816

max |119909 minus 119906|

119905 + |119909 minus 119906|

1003816100381610038161003816119910 minus V1003816100381610038161003816119905 +

1003816100381610038161003816119910 minus V1003816100381610038161003816)

ge119871lowastT (M

119872119873(119909 119906 119905) M

119872119873(119910 V 119905))

(97)




References




























Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in









1 1198871max119886

2 1198872) for all 119886 = (119886

1 1198862) and


M119872119873

(119909 119910 119905) = (119905

119905 +1003816100381610038161003816119909 minus 119910

1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119905 +1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)

forall119909 119910 isin 119883 119905 gt 0

(69)

Then (119883M119872119873


119892 (119909) =

0 119909 = 0

1 119909 =1

2119899 + 1

1

2119899 + 1 119909 =

1

2119899

119865 (119909 119910) =

1

(2119899 + 1)4 (119909 119910) = (

1

21198991

2119899)

0 otherwise

(70)

Let 119909119899= 119910119899= 12119899 We have

119892119909119899=

1

2119899 + 1997888rarr 0

119865 (119909119899 119910119899) =

1


(71)

but

M119872119873

(119865 (119892119909119899 119892119910119899) 119892119865 (119909

119899 119910119899) 119905)

= M119872119873

(0 1 119905) 999424999426999456 1119871lowast

(72)


119905

119883 + 119905ge min 119905

119884 + 119905

119905

119885 + 119905

119883

119883 + 119905le max 119884

119884 + 119905

119885

119885 + 119905


(73)



M119872119873

(119865 (119909 119910) 119865 (119906 V) 120601 (119905))

ge119871lowastT (M

119872119873(119892119909 119892119906 119905) M

119872119873(119892119910 119892V 119905))

(74)


21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)


1003816100381610038161003816 (75)





21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816 =2

(2119899 + 1)4

max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V

1003816100381610038161003816 =2119899

2119899 + 1

(76)




21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816 =2

(2119899 + 1)4

max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V

1003816100381610038161003816

= max 1

2119899 + 110038161003816100381610038161003816100381610038161003816

1

2119895 + 1minus

1

2119899 + 1

10038161003816100381610038161003816100381610038161003816

(77)

or

max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V

1003816100381610038161003816

= max 2119899

2119899 + 110038161003816100381610038161003816100381610038161003816

1

2119895 + 1minus

1

2119899 + 1

10038161003816100381610038161003816100381610038161003816

(78)

So (75) holds


21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816 =2

(2119899 + 1)4

max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V

1003816100381610038161003816

= max1003816100381610038161003816100381610038161003816

1

2119894 + 1minus

1

2119899 + 1

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1

2119895 + 1minus

1

2119899 + 1

10038161003816100381610038161003816100381610038161003816

(79)



21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816 = 2

100381610038161003816100381610038161003816100381610038161003816

1

(2119894 + 1)4minus

1

(2119899 + 1)4

100381610038161003816100381610038161003816100381610038161003816

max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V

1003816100381610038161003816

=1

2119894 + 1minus

1

2119899 + 1

(80)




119909 (119901) = int119887

119886

(1198701(119901 119902) + 119870

2(119901 119902))

times [119891 (119902 119909 (119902)) + 119892 (119902 119909 (119902))] 119889119902 + ℎ (119901)

(81)




(ii120579) 120579(119901) le 119901




1198701(119901 119902) ge 0 119870

2(119901 119902) le 0 forall119901 119902 isin 119868 (82)


0 le 119891 (119902 119909) minus 119891 (119902 119910) le 120582120579 (119909 minus 119910)


(iii) Consider

max 120582 120583 sup119901isin119868

int119887

119886

[1198701(119901 119902) minus 119870

2(119901 119902)] 119889119902 le

1

8 (84)






M119872119873

(119909 119910 119905) = (119905

119905 +1003816100381610038161003816119909 minus 119910

1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119905 +1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)

forall119909 119910 isin 119883 119905 gt 0

(85)


1 1198871max119886

2 1198872) for all 119886 = (119886

1 1198862) and 119887 = (119887

1 1198872) isin


119865 (119909 119910) (119901) = int119887

119886

1198701(119901 119902) [119891 (119902 119909 (119902)) + 119892 (119902 119910 (119902))] 119889119902

+ int119887

119886

1198702(119901 119902) [119891 (119902 119910 (119902))

+119892 (119902 119909 (119902))] 119889119902 + ℎ (119901)

(86)


119865 (119909 119910) (119901) minus 119865 (119906 V) (119901)

= int119887

119886

1198701(119901 119902) [119891 (119902 119909 (119902)) + 119892 (119902 119910 (119902))] 119889119902

+ int119887

119886

1198702(119901 119902) [119891 (119902 119910 (119902)) + 119892 (119902 119909 (119902))] 119889119902

minus int119887

119886

1198701(119901 119902) [119891 (119902 119906 (119902)) + 119892 (119902 V (119902))] 119889119902

minus int119887

119886

1198702(119901 119902) [119891 (119902 V (119902)) + 119892 (119902 119906 (119902))] 119889119902

= int119887

119886

1198701(119901 119902) [119891 (119902 119909 (119902)) minus 119891 (119902 119906 (119902))

+119892 (119902 119910 (119902)) minus 119892 (119902 V (119902))] 119889119902

+ int119887

119886

1198702(119901 119902) [119891 (119902 119910 (119902)) minus 119891 (119902 V (119902))

+119892 (119902 119909 (119902)) minus 119892 (119902 119906 (119902))] 119889119902

= int119887

119886

1198701(119901 119902) [(119891 (119902 119909 (119902)) minus 119891 (119902 119906 (119902)))

minus (119892 (119902 V (119902)) minus 119892 (119902 119910 (119902)))] 119889119902

minus int119887

119886

1198702(119901 119902) [(119891 (119902 V (119902)) minus 119891 (119902 119910 (119902)))

minus (119892 (119902 119909 (119902)) minus 119892 (119902 119906 (119902)))] 119889119902

le int119887

119886

1198701(119901 119902) [120582120579 (119909 (119902) minus 119906 (119902))

+120583120579 (V (119902) minus 119910 (119902))] 119889119902


minus int119887

119886

1198702(119901 119902) [120582120579 (V (119902) minus 119910 (119902))

+120583120579 (119909 (119902) minus 119906 (119902))] 119889119902

(87)

Thus

119865 (119909 119910) (119901) minus 119865 (119906 V) (119901)

le int119887

119886

1198701(119901 119902) [120582120579 (119909 (119902) minus 119906 (119902))

+120583120579 (V (119902) minus 119910 (119902))] 119889119902

minus int119887

119886

1198702(119901 119902) [120582120579 (V (119902) minus 119910 (119902))

+120583120579 (119909 (119902) minus 119906 (119902))] 119889119902

(88)


120579 (119909 (119902) minus 119906 (119902)) le 120579 (1003816100381610038161003816119909 (119902) minus 119906 (119902)

1003816100381610038161003816)

120579 (V (119902) minus 119910 (119902)) le 120579 (1003816100381610038161003816V (119902) minus 119910 (119902)

1003816100381610038161003816) (89)


1003816100381610038161003816119865 (119909 119910) (119901) minus 119865 (119906 V) (119901)1003816100381610038161003816

le int119887

119886

1198701(119901 119902) [120582120579 (

1003816100381610038161003816119909 (119902) minus 119906 (119902)1003816100381610038161003816)

+120583120579 (1003816100381610038161003816V (119902) minus 119910 (119902)

1003816100381610038161003816)] 119889119902

minus int119887

119886

1198702(119901 119902) [120582120579 (

1003816100381610038161003816V (119902) minus 119910 (119902)1003816100381610038161003816)

+120583120579 (1003816100381610038161003816119909 (119902) minus 119906 (119902)

1003816100381610038161003816)] 119889119902

le int119887

119886

1198701(119901 119902) [max 120582 120583 120579 (1003816100381610038161003816119909 (119902) minus 119906 (119902)

1003816100381610038161003816)

+max 120582 120583 120579 (1003816100381610038161003816V (119902) minus 119910 (119902)1003816100381610038161003816)] 119889119902

minus int119887

119886

1198702(119901 119902) [max 120582 120583 120579 (1003816100381610038161003816V (119902) minus 119910 (119902)

1003816100381610038161003816)

+max 120582 120583 120579 (1003816100381610038161003816119909 (119902) minus 119906 (119902)1003816100381610038161003816)] 119889119902

(90)


1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)1003816100381610038161003816

le max 120582 120583 int119887

119886

[1198701(119901 119902) minus 119870

2(119901 119902)] 119889119902

sdot [120579 (1003816100381610038161003816119909 (119902)minus119906 (119902)

1003816100381610038161003816)+120579 (1003816100381610038161003816V (119902) minus 119910 (119902)

1003816100381610038161003816)]

le max 120582 120583 sup119901isin119868

int119887

119886

[1198701(119901 119902) minus 119870

2(119901 119902)] 119889119902

sdot [120579 (1003816100381610038161003816119909 (119902)minus119906 (119902)

1003816100381610038161003816)+120579 (1003816100381610038161003816V (119902) minus 119910 (119902)

1003816100381610038161003816)]

le120579 (|119909 minus 119906|) + 120579 (

1003816100381610038161003816V minus 1199101003816100381610038161003816)

8

(91)

Thus

21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816 le120579 (|119909 minus 119906|) + 120579 (

1003816100381610038161003816V minus 1199101003816100381610038161003816)

4 (92)



120579 (1003816100381610038161003816119910 minus V1003816100381610038161003816) le 120579 (|119909 minus 119906| +

1003816100381610038161003816119910 minus V1003816100381610038161003816) (93)


120579 (|119909 minus 119906|) + 120579 (1003816100381610038161003816119910 minus V1003816100381610038161003816)

2le 120579 (|119909 minus 119906| +

1003816100381610038161003816119910 minus V1003816100381610038161003816)

le |119909 minus 119906| +1003816100381610038161003816119910 minus V1003816100381610038161003816


(94)

and so

120579 (|119909 minus 119906|) + 120579 (1003816100381610038161003816119910 minus V1003816100381610038161003816)



21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)



M119872119873

(119865 (119909 119910) 119865 (119906 V) 120601 (119905))

= M119872119873

(119865 (119909 119910) 119865 (119906 V) 119905

2)

= (1199052

1199052 +1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816

1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)1003816100381610038161003816

1199052 +1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816)

= (119905

119905 + 21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816

21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816119905 + 2

1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)1003816100381610038161003816)

ge119871lowast (

119905

119905 +max |119909 minus 119906| 1003816100381610038161003816119910 minus V1003816100381610038161003816


)



119905 + |119909 minus 119906|

119905

119905 +1003816100381610038161003816119910 minus V1003816100381610038161003816

max |119909 minus 119906|

119905 + |119909 minus 119906|

1003816100381610038161003816119910 minus V1003816100381610038161003816119905 +

1003816100381610038161003816119910 minus V1003816100381610038161003816)

ge119871lowastT (M

119872119873(119909 119906 119905) M

119872119873(119910 V 119905))

(97)




References




























Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in





21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816 = 2

100381610038161003816100381610038161003816100381610038161003816

1

(2119894 + 1)4minus

1

(2119899 + 1)4

100381610038161003816100381610038161003816100381610038161003816

max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V

1003816100381610038161003816

=1

2119894 + 1minus

1

2119899 + 1

(80)




119909 (119901) = int119887

119886

(1198701(119901 119902) + 119870

2(119901 119902))

times [119891 (119902 119909 (119902)) + 119892 (119902 119909 (119902))] 119889119902 + ℎ (119901)

(81)




(ii120579) 120579(119901) le 119901




1198701(119901 119902) ge 0 119870

2(119901 119902) le 0 forall119901 119902 isin 119868 (82)


0 le 119891 (119902 119909) minus 119891 (119902 119910) le 120582120579 (119909 minus 119910)


(iii) Consider

max 120582 120583 sup119901isin119868

int119887

119886

[1198701(119901 119902) minus 119870

2(119901 119902)] 119889119902 le

1

8 (84)






M119872119873

(119909 119910 119905) = (119905

119905 +1003816100381610038161003816119909 minus 119910

1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119905 +1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)

forall119909 119910 isin 119883 119905 gt 0

(85)


1 1198871max119886

2 1198872) for all 119886 = (119886

1 1198862) and 119887 = (119887

1 1198872) isin


119865 (119909 119910) (119901) = int119887

119886

1198701(119901 119902) [119891 (119902 119909 (119902)) + 119892 (119902 119910 (119902))] 119889119902

+ int119887

119886

1198702(119901 119902) [119891 (119902 119910 (119902))

+119892 (119902 119909 (119902))] 119889119902 + ℎ (119901)

(86)


119865 (119909 119910) (119901) minus 119865 (119906 V) (119901)

= int119887

119886

1198701(119901 119902) [119891 (119902 119909 (119902)) + 119892 (119902 119910 (119902))] 119889119902

+ int119887

119886

1198702(119901 119902) [119891 (119902 119910 (119902)) + 119892 (119902 119909 (119902))] 119889119902

minus int119887

119886

1198701(119901 119902) [119891 (119902 119906 (119902)) + 119892 (119902 V (119902))] 119889119902

minus int119887

119886

1198702(119901 119902) [119891 (119902 V (119902)) + 119892 (119902 119906 (119902))] 119889119902

= int119887

119886

1198701(119901 119902) [119891 (119902 119909 (119902)) minus 119891 (119902 119906 (119902))

+119892 (119902 119910 (119902)) minus 119892 (119902 V (119902))] 119889119902

+ int119887

119886

1198702(119901 119902) [119891 (119902 119910 (119902)) minus 119891 (119902 V (119902))

+119892 (119902 119909 (119902)) minus 119892 (119902 119906 (119902))] 119889119902

= int119887

119886

1198701(119901 119902) [(119891 (119902 119909 (119902)) minus 119891 (119902 119906 (119902)))

minus (119892 (119902 V (119902)) minus 119892 (119902 119910 (119902)))] 119889119902

minus int119887

119886

1198702(119901 119902) [(119891 (119902 V (119902)) minus 119891 (119902 119910 (119902)))

minus (119892 (119902 119909 (119902)) minus 119892 (119902 119906 (119902)))] 119889119902

le int119887

119886

1198701(119901 119902) [120582120579 (119909 (119902) minus 119906 (119902))

+120583120579 (V (119902) minus 119910 (119902))] 119889119902


minus int119887

119886

1198702(119901 119902) [120582120579 (V (119902) minus 119910 (119902))

+120583120579 (119909 (119902) minus 119906 (119902))] 119889119902

(87)

Thus

119865 (119909 119910) (119901) minus 119865 (119906 V) (119901)

le int119887

119886

1198701(119901 119902) [120582120579 (119909 (119902) minus 119906 (119902))

+120583120579 (V (119902) minus 119910 (119902))] 119889119902

minus int119887

119886

1198702(119901 119902) [120582120579 (V (119902) minus 119910 (119902))

+120583120579 (119909 (119902) minus 119906 (119902))] 119889119902

(88)


120579 (119909 (119902) minus 119906 (119902)) le 120579 (1003816100381610038161003816119909 (119902) minus 119906 (119902)

1003816100381610038161003816)

120579 (V (119902) minus 119910 (119902)) le 120579 (1003816100381610038161003816V (119902) minus 119910 (119902)

1003816100381610038161003816) (89)


1003816100381610038161003816119865 (119909 119910) (119901) minus 119865 (119906 V) (119901)1003816100381610038161003816

le int119887

119886

1198701(119901 119902) [120582120579 (

1003816100381610038161003816119909 (119902) minus 119906 (119902)1003816100381610038161003816)

+120583120579 (1003816100381610038161003816V (119902) minus 119910 (119902)

1003816100381610038161003816)] 119889119902

minus int119887

119886

1198702(119901 119902) [120582120579 (

1003816100381610038161003816V (119902) minus 119910 (119902)1003816100381610038161003816)

+120583120579 (1003816100381610038161003816119909 (119902) minus 119906 (119902)

1003816100381610038161003816)] 119889119902

le int119887

119886

1198701(119901 119902) [max 120582 120583 120579 (1003816100381610038161003816119909 (119902) minus 119906 (119902)

1003816100381610038161003816)

+max 120582 120583 120579 (1003816100381610038161003816V (119902) minus 119910 (119902)1003816100381610038161003816)] 119889119902

minus int119887

119886

1198702(119901 119902) [max 120582 120583 120579 (1003816100381610038161003816V (119902) minus 119910 (119902)

1003816100381610038161003816)

+max 120582 120583 120579 (1003816100381610038161003816119909 (119902) minus 119906 (119902)1003816100381610038161003816)] 119889119902

(90)


1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)1003816100381610038161003816

le max 120582 120583 int119887

119886

[1198701(119901 119902) minus 119870

2(119901 119902)] 119889119902

sdot [120579 (1003816100381610038161003816119909 (119902)minus119906 (119902)

1003816100381610038161003816)+120579 (1003816100381610038161003816V (119902) minus 119910 (119902)

1003816100381610038161003816)]

le max 120582 120583 sup119901isin119868

int119887

119886

[1198701(119901 119902) minus 119870

2(119901 119902)] 119889119902

sdot [120579 (1003816100381610038161003816119909 (119902)minus119906 (119902)

1003816100381610038161003816)+120579 (1003816100381610038161003816V (119902) minus 119910 (119902)

1003816100381610038161003816)]

le120579 (|119909 minus 119906|) + 120579 (

1003816100381610038161003816V minus 1199101003816100381610038161003816)

8

(91)

Thus

21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816 le120579 (|119909 minus 119906|) + 120579 (

1003816100381610038161003816V minus 1199101003816100381610038161003816)

4 (92)



120579 (1003816100381610038161003816119910 minus V1003816100381610038161003816) le 120579 (|119909 minus 119906| +

1003816100381610038161003816119910 minus V1003816100381610038161003816) (93)


120579 (|119909 minus 119906|) + 120579 (1003816100381610038161003816119910 minus V1003816100381610038161003816)

2le 120579 (|119909 minus 119906| +

1003816100381610038161003816119910 minus V1003816100381610038161003816)

le |119909 minus 119906| +1003816100381610038161003816119910 minus V1003816100381610038161003816


(94)

and so

120579 (|119909 minus 119906|) + 120579 (1003816100381610038161003816119910 minus V1003816100381610038161003816)



21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)



M119872119873

(119865 (119909 119910) 119865 (119906 V) 120601 (119905))

= M119872119873

(119865 (119909 119910) 119865 (119906 V) 119905

2)

= (1199052

1199052 +1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816

1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)1003816100381610038161003816

1199052 +1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816)

= (119905

119905 + 21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816

21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816119905 + 2

1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)1003816100381610038161003816)

ge119871lowast (

119905

119905 +max |119909 minus 119906| 1003816100381610038161003816119910 minus V1003816100381610038161003816


)



119905 + |119909 minus 119906|

119905

119905 +1003816100381610038161003816119910 minus V1003816100381610038161003816

max |119909 minus 119906|

119905 + |119909 minus 119906|

1003816100381610038161003816119910 minus V1003816100381610038161003816119905 +

1003816100381610038161003816119910 minus V1003816100381610038161003816)

ge119871lowastT (M

119872119873(119909 119906 119905) M

119872119873(119910 V 119905))

(97)




References




























Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




minus int119887

119886

1198702(119901 119902) [120582120579 (V (119902) minus 119910 (119902))

+120583120579 (119909 (119902) minus 119906 (119902))] 119889119902

(87)

Thus

119865 (119909 119910) (119901) minus 119865 (119906 V) (119901)

le int119887

119886

1198701(119901 119902) [120582120579 (119909 (119902) minus 119906 (119902))

+120583120579 (V (119902) minus 119910 (119902))] 119889119902

minus int119887

119886

1198702(119901 119902) [120582120579 (V (119902) minus 119910 (119902))

+120583120579 (119909 (119902) minus 119906 (119902))] 119889119902

(88)


120579 (119909 (119902) minus 119906 (119902)) le 120579 (1003816100381610038161003816119909 (119902) minus 119906 (119902)

1003816100381610038161003816)

120579 (V (119902) minus 119910 (119902)) le 120579 (1003816100381610038161003816V (119902) minus 119910 (119902)

1003816100381610038161003816) (89)


1003816100381610038161003816119865 (119909 119910) (119901) minus 119865 (119906 V) (119901)1003816100381610038161003816

le int119887

119886

1198701(119901 119902) [120582120579 (

1003816100381610038161003816119909 (119902) minus 119906 (119902)1003816100381610038161003816)

+120583120579 (1003816100381610038161003816V (119902) minus 119910 (119902)

1003816100381610038161003816)] 119889119902

minus int119887

119886

1198702(119901 119902) [120582120579 (

1003816100381610038161003816V (119902) minus 119910 (119902)1003816100381610038161003816)

+120583120579 (1003816100381610038161003816119909 (119902) minus 119906 (119902)

1003816100381610038161003816)] 119889119902

le int119887

119886

1198701(119901 119902) [max 120582 120583 120579 (1003816100381610038161003816119909 (119902) minus 119906 (119902)

1003816100381610038161003816)

+max 120582 120583 120579 (1003816100381610038161003816V (119902) minus 119910 (119902)1003816100381610038161003816)] 119889119902

minus int119887

119886

1198702(119901 119902) [max 120582 120583 120579 (1003816100381610038161003816V (119902) minus 119910 (119902)

1003816100381610038161003816)

+max 120582 120583 120579 (1003816100381610038161003816119909 (119902) minus 119906 (119902)1003816100381610038161003816)] 119889119902

(90)


1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)1003816100381610038161003816

le max 120582 120583 int119887

119886

[1198701(119901 119902) minus 119870

2(119901 119902)] 119889119902

sdot [120579 (1003816100381610038161003816119909 (119902)minus119906 (119902)

1003816100381610038161003816)+120579 (1003816100381610038161003816V (119902) minus 119910 (119902)

1003816100381610038161003816)]

le max 120582 120583 sup119901isin119868

int119887

119886

[1198701(119901 119902) minus 119870

2(119901 119902)] 119889119902

sdot [120579 (1003816100381610038161003816119909 (119902)minus119906 (119902)

1003816100381610038161003816)+120579 (1003816100381610038161003816V (119902) minus 119910 (119902)

1003816100381610038161003816)]

le120579 (|119909 minus 119906|) + 120579 (

1003816100381610038161003816V minus 1199101003816100381610038161003816)

8

(91)

Thus

21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816 le120579 (|119909 minus 119906|) + 120579 (

1003816100381610038161003816V minus 1199101003816100381610038161003816)

4 (92)



120579 (1003816100381610038161003816119910 minus V1003816100381610038161003816) le 120579 (|119909 minus 119906| +

1003816100381610038161003816119910 minus V1003816100381610038161003816) (93)


120579 (|119909 minus 119906|) + 120579 (1003816100381610038161003816119910 minus V1003816100381610038161003816)

2le 120579 (|119909 minus 119906| +

1003816100381610038161003816119910 minus V1003816100381610038161003816)

le |119909 minus 119906| +1003816100381610038161003816119910 minus V1003816100381610038161003816


(94)

and so

120579 (|119909 minus 119906|) + 120579 (1003816100381610038161003816119910 minus V1003816100381610038161003816)



21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)



M119872119873

(119865 (119909 119910) 119865 (119906 V) 120601 (119905))

= M119872119873

(119865 (119909 119910) 119865 (119906 V) 119905

2)

= (1199052

1199052 +1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816

1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)1003816100381610038161003816

1199052 +1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816)

= (119905

119905 + 21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816

21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)

1003816100381610038161003816119905 + 2

1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)1003816100381610038161003816)

ge119871lowast (

119905

119905 +max |119909 minus 119906| 1003816100381610038161003816119910 minus V1003816100381610038161003816


)



119905 + |119909 minus 119906|

119905

119905 +1003816100381610038161003816119910 minus V1003816100381610038161003816

max |119909 minus 119906|

119905 + |119909 minus 119906|

1003816100381610038161003816119910 minus V1003816100381610038161003816119905 +

1003816100381610038161003816119910 minus V1003816100381610038161003816)

ge119871lowastT (M

119872119873(119909 119906 119905) M

119872119873(119910 V 119905))

(97)




References




























Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in





119905 + |119909 minus 119906|

119905

119905 +1003816100381610038161003816119910 minus V1003816100381610038161003816

max |119909 minus 119906|

119905 + |119909 minus 119906|

1003816100381610038161003816119910 minus V1003816100381610038161003816119905 +

1003816100381610038161003816119910 minus V1003816100381610038161003816)

ge119871lowastT (M

119872119873(119909 119906 119905) M

119872119873(119910 V 119905))

(97)




References




























Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in










Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in



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Research Article Application of Coupled Fixed Point...

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