Hindawi Publishing CorporationJournal of MathematicsVolume 2013 Article ID 295093 8 pageshttpdxdoiorg1011552013295093
Research ArticleSome Fixed Point Theorems for PrešiT-Hardy-Rogers TypeContractions in Metric Spaces
Satish Shukla1 Stojan RadenoviT2 and Slaviša PanteliT2
1 Department of Applied Mathematics Shri Vaishnav Institute of Technology and Science Gram Baroli Sanwer RoadIndore 453331 India
2 Faculty of Mechanical Engineering University of Belgrade Kraljice Marije 16 11120 Beograd Serbia
Correspondence should be addressed to Satish Shukla satishmathematicsyahoocoin
Received 2 January 2013 Accepted 7 February 2013
Academic Editor NanJing Huang
Copyright copy 2013 Satish Shukla et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
We introduce some generalizations of Presic type contractions and establish some fixed point theorems for mappings satisfyingPresic-Hardy-Rogers type contractive conditions inmetric spaces Our results generalize and extend several known results inmetricspaces Some examples are included which illustrate the cases when new results can be applied while old ones cannot
1 Introduction
Thewell-knownBanach contractionmapping principle statesthat if (119883 119889) is a complete metric space and 119879 119883 rarr 119883 is aself-mapping such that
119889 (119879119909 119879119910) le 120582119889 (119909 119910) (1)
for all 119909 119910 isin 119883 where 0 le 120582 lt 1 then there exists a unique119909 isin 119883 such that 119879119909 = 119909 This point 119909 is called the fixed pointof mapping 119879
On the other hand for mappings 119879 119883 rarr 119883 Kannan[1] introduced the contractive condition
for all 119909 119910 isin 119883 where 120582 isin [0 12) is a constant and proveda fixed point theorem using (2) instead of (1) The conditions(1) and (2) are independent as it was shown by two examplesin [2]
Reich [3] for mappings 119879 119883 rarr 119883 generalizedBanach and Kannan fixed point theorems using contractivecondition
for all 119909 119910 isin 119883 where 120572 120573 120574 are nonnegative constants with120572 + 120573 + 120574 lt 1 An example in [3] shows that the condition (3)is a proper generalization of (1) and (2)
for all 119909 119910 isin 119883 where 120572 120573 120574 120575 are nonnegative constantswith 120572 + 120573 + 120574 + 2120575 lt 1 A mapping satisfying (5) is calledGeneralized contraction
Hardy and Rogers [6] for mappings 119879 119883 rarr 119883 usedthe contractive condition
for all 119909 119910 isin 119883 where 120572 120573 120574 120575 120583 are nonnegative constantswith 120572+120573+ 120574 + 120575 + 120583 lt 1 and proved fixed point result Notethat condition (6) generalizes all the previous conditions
2 Journal of Mathematics
In 1965 Presic [7 8] extended Banach contraction map-ping principle to mappings defined on product spaces andproved the following theorem
Theorem 1 Let (119883 119889) be a complete metric space 119896 a positiveinteger and 119891 119883119896 rarr 119883 a mapping satisfying the followingcontractive type condition
for every 1199091 1199092 119909119896+1 isin 119883 where 1199021 1199022 119902119896 are non-negative constants such that 1199021 + 1199022 + sdot sdot sdot + 119902119896 lt 1 Then thereexists a unique point 119909 isin 119883 such that 119891(119909 119909 119909) = 119909Moreover if 1199091 1199092 119909119896 are arbitrary points in 119883 and for119899 isin N
then the sequence 119909119899 is convergent and lim119909119899 = 119891(lim119909119899lim119909119899 lim119909119899)
Note that condition (7) in the case 119896 = 1 reducesto the well-known Banach contraction mapping principleSo Theorem 1 is a generalization of the Banach fixed pointtheorem Some generalizations and applications of Presictheorem can be seen in [9ndash18]
The 119896-step iterative sequence given by (8) represents anonlinear difference equation and the solution of this equa-tion can be assumed to be a fixed point of 119891 that is solutionof (8) is a point 119909lowast isin 119883 such that 119909lowast = 119891(119909lowast 119909lowast 119909lowast)The Presic theorem insures the convergence of the sequence119909119899 defined by (8) and provides a sufficient condition forthe existence of solution of (8) in the case when mapping 119891satisfies the condition (7) A condition independent from (7)namely the Presic-Kannan condition is considered in [11](for the proof of independency of these conditions in case119896 = 1 we refer [1 2]) In this paper we introduce somegeneralizations of Presic type contractions in metric spacesand use a more general condition namely the Presic-Hardy-Rogers type condition to prove the existence of fixed pointof 119891 in metric spaces We note that this condition generalizesthe result of Presic [7 8] Pacurar [11] Hardy and Rogers [6]and several known results in metric spaces Some examplesare included which illustrate the cases when new results canbe applied while old ones cannot
2 Some Generalizations ofPresiT Type Contractions
In this section we introduced some Presic type contractionsin metric spaces
Let (119883 119889) be a metric space 119896 a positive integer and 119891 119883119896 rarr 119883 be a mapping
(i) 119891 is said to be a Presic contraction if 119891 satisfies thecondition (7)
(ii) 119891 is said to be a Presic-Kannan contraction (see [11]for detail) if 119891 satisfies following condition
for all 1199091 1199092 119909119896 119909119896+1 isin 119883 where 120572119894 120573119894119895 are non-negative constants such that
119896
sum119894=1
120572119894 + 119896
119896+1
sum119894=1
119896+1
sum119895=1
120573119894119895 lt 1 (18)
Remark 2 Note that for 120573119894119895 = 120573 for all 119894 119895 isin 1 2 119896 119896+1with 119894 = 119895 and 120573119894119894 = 120573119894 for all 119894 isin 1 2 119896 119896+1 the Presic-Hardy-Rogers contraction reduces into the Generalized-Presic contraction With 120573 = 0 the Generalized-Presiccontraction reduces into the Presic-Reich contraction andwith 120572119894 = 0 for all 119894 isin 1 2 119896 120573119894 = 0 for all 119894 isin1 2 119896 119896 + 1 and 120573 = 120574 the Generalized-Presic con-traction reduces into the Presic-Chatterjea contraction With120572119894 = 0 for all 119894 isin 1 2 119896 the Presic-Reich contractionreduces into the Presic-Kannan contraction and with 120573119894 = 0for all 119894 isin 1 2 119896 119896 + 1 the Presic-Reich contractionreduces into the Presic contraction Therefore among allabove definitions the Presic-Hardy-Rogers contraction is themost general contraction
Remark 3 It is easy to see that for 119896 = 1 Presic-Hardy-Rogers contraction reduces into Hardy-Rogers contractionand for 119896 = 1 Generalized-Presic contraction reducesinto Generalized contraction and so forth therefore thecomparison as considered in [19] shows that the abovegeneralization is proper
Now we shall prove some fixed point results for Presic-Hardy-Rogers type contractions in metric spaces
3 Main Results
The following theorem is the fixed point result for Presic-Hardy-Rogers type contractions and the main result of thispaper
Theorem 4 Let (119883 119889) be any complete metric space 119896 apositive integer Let 119891 119883119896 rarr 119883 be a Presic-Hardy-Rogerscontraction then 119891 has a unique fixed point in119883
Proof Let 1199090 isin 119883 be arbitrary Define a sequence 119909119899 in 119883by
119909119899+1 = 119891 (119909119899 119909119899) 119899 ge 0 (19)
If 119909119899 = 119909119899+1 for any 119899 then 119909119899 is a fixed point of 119891 Thereforewe assume 119909119899 = 119909119899+1 for all 119899
We shall show that this sequence is a Cauchy sequence in119883
where119860 119861 119862 119864 and 119865 are the coefficients of 119889119899 119863119899119899 119863119899119899+1119863119899+1119899 and119863119899+1119899+1 respectively in the above inequality
(1 minus 119862 minus 119865) 119889119899+1 le (119860 + 119861 + 119862) 119889119899 (26)
Again as 119889119899+1 = 119889(119909119899 119909119899+1) = 119889(119909119899+1 119909119899) interchanging therole of 119909119899 and 119909119899+1 and repeating above process we obtain
(1 minus 119864 minus 119861) 119889119899+1 le (119860 + 119865 + 119864) 119889119899 (27)
It follows from (26) and (27) that(2 minus 119861 minus 119862 minus 119864 minus 119865) 119889119899+1 le (2119860 + 119861 + 119862 + 119864 + 119865) 119889119899
as 0 le 120582 lt 1 it follows from the above inequalitythat lim119899rarrinfin119889(119909119899 119909119898) = 0 Therefore 119909119899 is a Cauchysequence By completeness of119883 there exists 119906 isin 119883 such thatlim119899rarrinfin119909119899 = 119906
We shall show that 119906 is the fixed point of 119891 Note that
119889 (119906 119891 (119906 119906))
le 119889 (119906 119909119899+1) + 119889 (119909119899+1 119891 (119906 119906))
1 minus 119862 minus 119865119889 (119909119899 119906) +
1 + 119861 + 119864
1 minus 119862 minus 119865119889 (119909119899+1 119906)
(34)
Using the fact that lim119899rarrinfin119909119899 = 119906 it follows from the aboveinequality that
119889 (119906 119891 (119906 119906)) = 0 that is 119891 (119906 119906) = 119906 (35)
Thus 119906 is a fixed point of 119891 For uniqueness let V be anotherfixed point of 119891 that is 119891(V V) = V Again using a similarprocess as used in the calculation of 119889119899+1 we obtain
+ 119862119889 (119906 119891 (V V)) + 119864119889 (V 119891 (119906 119906))
+ 119865119889 (V 119891 (V V))
= (119860 + 119862 + 119864) 119889 (119906 V)
(36)
as 119860+119861 +119862 + 119864 + 119865 lt 1 we obtain 119889(119906 V) = 0 that is 119906 = VThus fixed point is unique
Remark 5 For 119896 = 1 in the above theorem we obtain theresult of Hardy and Rogers [6] For 120573119894119895 = 0 for all 119894 119895 isin1 2 119896 119896 + 1 we obtain the fixed point result of PresicTherefore above theorem is a generalization of the results ofHardy and Rogers and Presic
With Remark 2 the following corollaries are obtained
Corollary 6 Let (119883 119889) be any complete metric space 119896 apositive integer and 119891 119883119896 rarr 119883 a Generalized Presiccontraction Then 119891 has a unique fixed point in119883
For 119896 = 1 in above corollary we obtain the fixed pointresult of Ciric [5]
Corollary 7 Let (119883 119889) be any complete metric space 119896 apositive integer and 119891 119883119896 rarr 119883 a Presic-Reich contractionThen 119891 has a unique fixed point in119883
For 119896 = 1 in the above corollary we obtain the fixed pointresult of Reich [3]
Corollary 8 Let (119883 119889) be any complete metric space 119896 apositive integer and 119891 119883119896 rarr 119883 a Presic-Kannancontraction Then 119891 has a unique fixed point in119883
For 119896 = 1 in above the corollary we obtain the fixed pointresult of Kannan [2]
Corollary 9 Let (119883 119889) be any complete metric space 119896 apositive integer and 119891 119883119896 rarr 119883 a Presic-Chatterjeacontraction Then 119891 has a unique fixed point in119883
For 119896 = 1 in above corollary we obtain the fixed pointresult of Chatterjea [4]
The following are some examples which illustrate thecases when known results are not applicable while our newresults can be used to conclude the existence of fixed point ofmapping
6 Journal of Mathematics
Example 10 Let119883 = [0 1]with usualmetric For 119896 = 2define119891 1198832 rarr 119883 by
119891 (119909 119910) =
1
5 if 119909 = 119910 = 1
119909 + 119910
5 otherwise
(37)
Then
(i) 119891 is a Presic-Reich contraction with 1205721 = 1205722 =
15 1205731 = 1205732 = 1205733 = 111(ii) 119891 is not a Presic contraction(iii) 119891 is not a Presic-Kannan contraction
Proof (i) Note that for 1199091 1199092 1199093 isin [0 1) with 1199091 le 1199092 le 1199093
Therefore conditions (11) and (12) are satisfied for 1205721 = 1205722 =15 and 1205731 1205732 1205733 with 1205731 + 1205732 + 1205733 isin [0 310)
If any one of 1199091 1199092 1199093 is 1 then proof is similar If any twoof 1199091 1199092 1199093 are 1 for example if 1199091 isin [0 1) and 1199092 = 1199093 = 1then
As 1199091 isin [0 1) so conditions (11) and (12) are satisfied for 1205731 =1205732 = 1205733 = 111 and 1205721 1205722 with 1205721 + 1205722 isin [0 511)
Similarly in all possible cases conditions (11) and (12)are satisfied with 1205721 = 1205722 = 15 1205731 = 1205732 = 1205733 =
111 Therefore 119891 is a Presic-Reich contraction All otherconditions of Corollary 7 are satisfied and 0 is the uniquefixed point of 119891
(ii) Note that for 1199091 = 910 and 1199092 = 1199093 = 1
Therefore we cannot find nonnegative constants 1205721 1205722 suchthat condition (7) is satisfied with 1205721 + 1205722 lt 1 So 119891 is not aPresic contraction
Therefore we cannot find nonnegative constant 120573 such thatconditions (9) and (10) are satisfied So 119891 is not a Presic-Kannan contraction
Remark 11 In the above example we cannot apply the resultof Presic [7 8] and Pacurar [11] to conclude the existence offixed point of 119891 But Corollary 7 is applicable which insuresthe existence of unique fixed point of 119891
Example 12 Let 119883 = [0 1] with usual metric For 119896 = 2define 119891 1198832 rarr 119883 by
119891 (119909 119910) =
4
15 if 119909 = 119910 = 1
0 otherwise(42)
Then
(i) 119891 is a Presic-Chatterjea contraction with 120574 isin
[113 112)
(ii) 119891 is not a Presic contraction
(iii) 119891 is not a Presic-Kannan contraction
Proof (i)Note that if1199091 1199092 1199093 isin [0 1) or any one of1199091 1199092 1199093is 1 then conditions (17) and (18) are satisfied trivially
Journal of Mathematics 7
If any two of 1199091 1199092 1199093 are 1 for example if 1199091 isin [0 1)1199092 = 1199093 = 1 then
Therefore we cannot find nonnegative constants 1205721 1205722 suchthat condition (7) is satisfied with 1205721 + 1205722 lt 1 So 119891 is not aPresic contraction
(iii) For 1199091 = 0 1199092 = 1199093 = 1 we have
Therefore we cannot find nonnegative constant 120573 such thatconditions (9) and (10) are satisfied So 119891 is not a Presic-Kannan contraction
Remark 13 In the above example we cannot apply the resultof Presic [7 8] and Pacurar [11] to conclude the existence offixed point of 119891 But Corollary 9 is applicable which insuresthe existence of unique fixed point of 119891
The following theorem is a consequence of Theorem 4and the recent result of Aydi et al [20]
Theorem 14 Let (119883 119889) be any complete metric space and 119896 apositive integer Let 119891 119883119896 rarr 119883 and 119879 119883 rarr 119883 be twomappings such that the following condition holds
Then (119883 120588) is a complete metric space (see [20]) Note thatcondition (46) reduces to the condition (17) that is mapping119891 reduces to Presic-Hardy-Rogers contractionwith respect tometric 120588 So the rest of the proof followedTheorem 4
Acknowledgment
The first author is thankful to Mr Rajpal Tomar for his helpin typing the manuscript
References
[1] R Kannan ldquoSome results on fixed pointsrdquo Bulletin of theCalcutta Mathematical Society vol 60 pp 71ndash76 1968
[2] R Kannan ldquoSome results on fixed points IIrdquo The AmericanMathematical Monthly vol 76 pp 405ndash408 1969
[4] S K Chatterjea ldquoFixed-point theoremsrdquo Comptes Rendus delrsquoAcademie Bulgare des Sciences vol 25 pp 727ndash730 1972
[5] L B Ciric ldquoGeneralized contractions and fixed-point theo-remsrdquo Publications de lrsquoInstitut Mathematique vol 12 no 26pp 19ndash26 1971
8 Journal of Mathematics
[6] G E Hardy and T D Rogers ldquoA generalization of a fixed pointtheorem of Reichrdquo Canadian Mathematical Bulletin vol 16 pp201ndash206 1973
[7] S B Presic ldquoSur une classe drsquoinequations aux differencesfinies et sur la convergence de certaines suitesrdquo Publications delrsquoInstitut Mathematique vol 5 no 19 pp 75ndash78 1965
[8] S B Presic ldquoSur la convergence des suitesrdquo Comptes Rendus delrsquoAcademie des Sciences vol 260 pp 3828ndash3830 1965
[9] L B Ciric and S B Presic ldquoOn Presic type generalization ofthe Banach contraction mapping principlerdquo Acta MathematicaUniversitatis Comenianae vol 76 no 2 pp 143ndash147 2007
[10] M Pacurar ldquoA multi-step iterative method for approximatingcommon fixed points of Presic-Rus type operators on metricspacesrdquo Studia Universitatis Babes-Bolyai Mathematica vol 55no 1 p 149 2010
[11] M Pacurar ldquoApproximating common fixed points of Presic-Kannan type operators by a multi-step iterative methodrdquoAnalele Stiintifice ale Universitatii Ovidius Constanta SeriaMatematica vol 17 no 1 pp 153ndash168 2009
[12] M Pacurar ldquoCommon fixed points for almost Presic typeoperatorsrdquoCarpathian Journal of Mathematics vol 28 no 1 pp117ndash126 2012
[13] M S KhanM Berzig and B Samet ldquoSome convergence resultsfor iterative sequences of Presic type and applicationsrdquoAdvancesin Difference Equations vol 2012 article 38 2012
[14] R George K P Reshma and R Rajagopalan ldquoA generalisedfixed point theorem of Presic type in cone metric spacesand application to Markov processrdquo Fixed Point Theory andApplications vol 2011 article 85 2011
[15] S Shukla R Sen and S Radenovic ldquoSet-valued Presic typecontraction inmetric spacesrdquoAnalele Stiintifice ale UniversitatiiIn press
[16] S Shukla andR Sen ldquoSet-valued Presic-Reich typemappings inmetric spacesrdquo Revista de la Real Academia de Ciencias ExactasFisicas y Naturales A 2012
[17] S K Malhotra S Shukla and R Sen ldquoA generalization ofBanach contraction principle in ordered cone metric spacesrdquoJournal of Advanced Mathematical Studies vol 5 no 2 pp 59ndash67 2012
[18] Y-Z Chen ldquoA Presic type contractive condition and its applica-tionsrdquo Nonlinear Analysis vol 71 no 12 pp e2012ndashe2017 2009
[19] B E Rhoades ldquoA comparison of various definitions of con-tractive mappingsrdquo Transactions of the American MathematicalSociety vol 226 pp 257ndash290 1977
[20] H Aydi E Karapınar and B Samet ldquoRemarks on some recentfixed point theoremsrdquo Fixed Point Theory and Applications vol2012 article 76 2012
In 1965 Presic [7 8] extended Banach contraction map-ping principle to mappings defined on product spaces andproved the following theorem
Theorem 1 Let (119883 119889) be a complete metric space 119896 a positiveinteger and 119891 119883119896 rarr 119883 a mapping satisfying the followingcontractive type condition
for every 1199091 1199092 119909119896+1 isin 119883 where 1199021 1199022 119902119896 are non-negative constants such that 1199021 + 1199022 + sdot sdot sdot + 119902119896 lt 1 Then thereexists a unique point 119909 isin 119883 such that 119891(119909 119909 119909) = 119909Moreover if 1199091 1199092 119909119896 are arbitrary points in 119883 and for119899 isin N
then the sequence 119909119899 is convergent and lim119909119899 = 119891(lim119909119899lim119909119899 lim119909119899)
Note that condition (7) in the case 119896 = 1 reducesto the well-known Banach contraction mapping principleSo Theorem 1 is a generalization of the Banach fixed pointtheorem Some generalizations and applications of Presictheorem can be seen in [9ndash18]
The 119896-step iterative sequence given by (8) represents anonlinear difference equation and the solution of this equa-tion can be assumed to be a fixed point of 119891 that is solutionof (8) is a point 119909lowast isin 119883 such that 119909lowast = 119891(119909lowast 119909lowast 119909lowast)The Presic theorem insures the convergence of the sequence119909119899 defined by (8) and provides a sufficient condition forthe existence of solution of (8) in the case when mapping 119891satisfies the condition (7) A condition independent from (7)namely the Presic-Kannan condition is considered in [11](for the proof of independency of these conditions in case119896 = 1 we refer [1 2]) In this paper we introduce somegeneralizations of Presic type contractions in metric spacesand use a more general condition namely the Presic-Hardy-Rogers type condition to prove the existence of fixed pointof 119891 in metric spaces We note that this condition generalizesthe result of Presic [7 8] Pacurar [11] Hardy and Rogers [6]and several known results in metric spaces Some examplesare included which illustrate the cases when new results canbe applied while old ones cannot
2 Some Generalizations ofPresiT Type Contractions
In this section we introduced some Presic type contractionsin metric spaces
Let (119883 119889) be a metric space 119896 a positive integer and 119891 119883119896 rarr 119883 be a mapping
(i) 119891 is said to be a Presic contraction if 119891 satisfies thecondition (7)
(ii) 119891 is said to be a Presic-Kannan contraction (see [11]for detail) if 119891 satisfies following condition
for all 1199091 1199092 119909119896 119909119896+1 isin 119883 where 120572119894 120573119894119895 are non-negative constants such that
119896
sum119894=1
120572119894 + 119896
119896+1
sum119894=1
119896+1
sum119895=1
120573119894119895 lt 1 (18)
Remark 2 Note that for 120573119894119895 = 120573 for all 119894 119895 isin 1 2 119896 119896+1with 119894 = 119895 and 120573119894119894 = 120573119894 for all 119894 isin 1 2 119896 119896+1 the Presic-Hardy-Rogers contraction reduces into the Generalized-Presic contraction With 120573 = 0 the Generalized-Presiccontraction reduces into the Presic-Reich contraction andwith 120572119894 = 0 for all 119894 isin 1 2 119896 120573119894 = 0 for all 119894 isin1 2 119896 119896 + 1 and 120573 = 120574 the Generalized-Presic con-traction reduces into the Presic-Chatterjea contraction With120572119894 = 0 for all 119894 isin 1 2 119896 the Presic-Reich contractionreduces into the Presic-Kannan contraction and with 120573119894 = 0for all 119894 isin 1 2 119896 119896 + 1 the Presic-Reich contractionreduces into the Presic contraction Therefore among allabove definitions the Presic-Hardy-Rogers contraction is themost general contraction
Remark 3 It is easy to see that for 119896 = 1 Presic-Hardy-Rogers contraction reduces into Hardy-Rogers contractionand for 119896 = 1 Generalized-Presic contraction reducesinto Generalized contraction and so forth therefore thecomparison as considered in [19] shows that the abovegeneralization is proper
Now we shall prove some fixed point results for Presic-Hardy-Rogers type contractions in metric spaces
3 Main Results
The following theorem is the fixed point result for Presic-Hardy-Rogers type contractions and the main result of thispaper
Theorem 4 Let (119883 119889) be any complete metric space 119896 apositive integer Let 119891 119883119896 rarr 119883 be a Presic-Hardy-Rogerscontraction then 119891 has a unique fixed point in119883
Proof Let 1199090 isin 119883 be arbitrary Define a sequence 119909119899 in 119883by
119909119899+1 = 119891 (119909119899 119909119899) 119899 ge 0 (19)
If 119909119899 = 119909119899+1 for any 119899 then 119909119899 is a fixed point of 119891 Thereforewe assume 119909119899 = 119909119899+1 for all 119899
We shall show that this sequence is a Cauchy sequence in119883
where119860 119861 119862 119864 and 119865 are the coefficients of 119889119899 119863119899119899 119863119899119899+1119863119899+1119899 and119863119899+1119899+1 respectively in the above inequality
(1 minus 119862 minus 119865) 119889119899+1 le (119860 + 119861 + 119862) 119889119899 (26)
Again as 119889119899+1 = 119889(119909119899 119909119899+1) = 119889(119909119899+1 119909119899) interchanging therole of 119909119899 and 119909119899+1 and repeating above process we obtain
(1 minus 119864 minus 119861) 119889119899+1 le (119860 + 119865 + 119864) 119889119899 (27)
It follows from (26) and (27) that(2 minus 119861 minus 119862 minus 119864 minus 119865) 119889119899+1 le (2119860 + 119861 + 119862 + 119864 + 119865) 119889119899
as 0 le 120582 lt 1 it follows from the above inequalitythat lim119899rarrinfin119889(119909119899 119909119898) = 0 Therefore 119909119899 is a Cauchysequence By completeness of119883 there exists 119906 isin 119883 such thatlim119899rarrinfin119909119899 = 119906
We shall show that 119906 is the fixed point of 119891 Note that
119889 (119906 119891 (119906 119906))
le 119889 (119906 119909119899+1) + 119889 (119909119899+1 119891 (119906 119906))
1 minus 119862 minus 119865119889 (119909119899 119906) +
1 + 119861 + 119864
1 minus 119862 minus 119865119889 (119909119899+1 119906)
(34)
Using the fact that lim119899rarrinfin119909119899 = 119906 it follows from the aboveinequality that
119889 (119906 119891 (119906 119906)) = 0 that is 119891 (119906 119906) = 119906 (35)
Thus 119906 is a fixed point of 119891 For uniqueness let V be anotherfixed point of 119891 that is 119891(V V) = V Again using a similarprocess as used in the calculation of 119889119899+1 we obtain
+ 119862119889 (119906 119891 (V V)) + 119864119889 (V 119891 (119906 119906))
+ 119865119889 (V 119891 (V V))
= (119860 + 119862 + 119864) 119889 (119906 V)
(36)
as 119860+119861 +119862 + 119864 + 119865 lt 1 we obtain 119889(119906 V) = 0 that is 119906 = VThus fixed point is unique
Remark 5 For 119896 = 1 in the above theorem we obtain theresult of Hardy and Rogers [6] For 120573119894119895 = 0 for all 119894 119895 isin1 2 119896 119896 + 1 we obtain the fixed point result of PresicTherefore above theorem is a generalization of the results ofHardy and Rogers and Presic
With Remark 2 the following corollaries are obtained
Corollary 6 Let (119883 119889) be any complete metric space 119896 apositive integer and 119891 119883119896 rarr 119883 a Generalized Presiccontraction Then 119891 has a unique fixed point in119883
For 119896 = 1 in above corollary we obtain the fixed pointresult of Ciric [5]
Corollary 7 Let (119883 119889) be any complete metric space 119896 apositive integer and 119891 119883119896 rarr 119883 a Presic-Reich contractionThen 119891 has a unique fixed point in119883
For 119896 = 1 in the above corollary we obtain the fixed pointresult of Reich [3]
Corollary 8 Let (119883 119889) be any complete metric space 119896 apositive integer and 119891 119883119896 rarr 119883 a Presic-Kannancontraction Then 119891 has a unique fixed point in119883
For 119896 = 1 in above the corollary we obtain the fixed pointresult of Kannan [2]
Corollary 9 Let (119883 119889) be any complete metric space 119896 apositive integer and 119891 119883119896 rarr 119883 a Presic-Chatterjeacontraction Then 119891 has a unique fixed point in119883
For 119896 = 1 in above corollary we obtain the fixed pointresult of Chatterjea [4]
The following are some examples which illustrate thecases when known results are not applicable while our newresults can be used to conclude the existence of fixed point ofmapping
6 Journal of Mathematics
Example 10 Let119883 = [0 1]with usualmetric For 119896 = 2define119891 1198832 rarr 119883 by
119891 (119909 119910) =
1
5 if 119909 = 119910 = 1
119909 + 119910
5 otherwise
(37)
Then
(i) 119891 is a Presic-Reich contraction with 1205721 = 1205722 =
15 1205731 = 1205732 = 1205733 = 111(ii) 119891 is not a Presic contraction(iii) 119891 is not a Presic-Kannan contraction
Proof (i) Note that for 1199091 1199092 1199093 isin [0 1) with 1199091 le 1199092 le 1199093
Therefore conditions (11) and (12) are satisfied for 1205721 = 1205722 =15 and 1205731 1205732 1205733 with 1205731 + 1205732 + 1205733 isin [0 310)
If any one of 1199091 1199092 1199093 is 1 then proof is similar If any twoof 1199091 1199092 1199093 are 1 for example if 1199091 isin [0 1) and 1199092 = 1199093 = 1then
As 1199091 isin [0 1) so conditions (11) and (12) are satisfied for 1205731 =1205732 = 1205733 = 111 and 1205721 1205722 with 1205721 + 1205722 isin [0 511)
Similarly in all possible cases conditions (11) and (12)are satisfied with 1205721 = 1205722 = 15 1205731 = 1205732 = 1205733 =
111 Therefore 119891 is a Presic-Reich contraction All otherconditions of Corollary 7 are satisfied and 0 is the uniquefixed point of 119891
(ii) Note that for 1199091 = 910 and 1199092 = 1199093 = 1
Therefore we cannot find nonnegative constants 1205721 1205722 suchthat condition (7) is satisfied with 1205721 + 1205722 lt 1 So 119891 is not aPresic contraction
Therefore we cannot find nonnegative constant 120573 such thatconditions (9) and (10) are satisfied So 119891 is not a Presic-Kannan contraction
Remark 11 In the above example we cannot apply the resultof Presic [7 8] and Pacurar [11] to conclude the existence offixed point of 119891 But Corollary 7 is applicable which insuresthe existence of unique fixed point of 119891
Example 12 Let 119883 = [0 1] with usual metric For 119896 = 2define 119891 1198832 rarr 119883 by
119891 (119909 119910) =
4
15 if 119909 = 119910 = 1
0 otherwise(42)
Then
(i) 119891 is a Presic-Chatterjea contraction with 120574 isin
[113 112)
(ii) 119891 is not a Presic contraction
(iii) 119891 is not a Presic-Kannan contraction
Proof (i)Note that if1199091 1199092 1199093 isin [0 1) or any one of1199091 1199092 1199093is 1 then conditions (17) and (18) are satisfied trivially
Journal of Mathematics 7
If any two of 1199091 1199092 1199093 are 1 for example if 1199091 isin [0 1)1199092 = 1199093 = 1 then
Therefore we cannot find nonnegative constants 1205721 1205722 suchthat condition (7) is satisfied with 1205721 + 1205722 lt 1 So 119891 is not aPresic contraction
(iii) For 1199091 = 0 1199092 = 1199093 = 1 we have
Therefore we cannot find nonnegative constant 120573 such thatconditions (9) and (10) are satisfied So 119891 is not a Presic-Kannan contraction
Remark 13 In the above example we cannot apply the resultof Presic [7 8] and Pacurar [11] to conclude the existence offixed point of 119891 But Corollary 9 is applicable which insuresthe existence of unique fixed point of 119891
The following theorem is a consequence of Theorem 4and the recent result of Aydi et al [20]
Theorem 14 Let (119883 119889) be any complete metric space and 119896 apositive integer Let 119891 119883119896 rarr 119883 and 119879 119883 rarr 119883 be twomappings such that the following condition holds
Then (119883 120588) is a complete metric space (see [20]) Note thatcondition (46) reduces to the condition (17) that is mapping119891 reduces to Presic-Hardy-Rogers contractionwith respect tometric 120588 So the rest of the proof followedTheorem 4
Acknowledgment
The first author is thankful to Mr Rajpal Tomar for his helpin typing the manuscript
References
[1] R Kannan ldquoSome results on fixed pointsrdquo Bulletin of theCalcutta Mathematical Society vol 60 pp 71ndash76 1968
[2] R Kannan ldquoSome results on fixed points IIrdquo The AmericanMathematical Monthly vol 76 pp 405ndash408 1969
[4] S K Chatterjea ldquoFixed-point theoremsrdquo Comptes Rendus delrsquoAcademie Bulgare des Sciences vol 25 pp 727ndash730 1972
[5] L B Ciric ldquoGeneralized contractions and fixed-point theo-remsrdquo Publications de lrsquoInstitut Mathematique vol 12 no 26pp 19ndash26 1971
8 Journal of Mathematics
[6] G E Hardy and T D Rogers ldquoA generalization of a fixed pointtheorem of Reichrdquo Canadian Mathematical Bulletin vol 16 pp201ndash206 1973
[7] S B Presic ldquoSur une classe drsquoinequations aux differencesfinies et sur la convergence de certaines suitesrdquo Publications delrsquoInstitut Mathematique vol 5 no 19 pp 75ndash78 1965
[8] S B Presic ldquoSur la convergence des suitesrdquo Comptes Rendus delrsquoAcademie des Sciences vol 260 pp 3828ndash3830 1965
[9] L B Ciric and S B Presic ldquoOn Presic type generalization ofthe Banach contraction mapping principlerdquo Acta MathematicaUniversitatis Comenianae vol 76 no 2 pp 143ndash147 2007
[10] M Pacurar ldquoA multi-step iterative method for approximatingcommon fixed points of Presic-Rus type operators on metricspacesrdquo Studia Universitatis Babes-Bolyai Mathematica vol 55no 1 p 149 2010
[11] M Pacurar ldquoApproximating common fixed points of Presic-Kannan type operators by a multi-step iterative methodrdquoAnalele Stiintifice ale Universitatii Ovidius Constanta SeriaMatematica vol 17 no 1 pp 153ndash168 2009
[12] M Pacurar ldquoCommon fixed points for almost Presic typeoperatorsrdquoCarpathian Journal of Mathematics vol 28 no 1 pp117ndash126 2012
[13] M S KhanM Berzig and B Samet ldquoSome convergence resultsfor iterative sequences of Presic type and applicationsrdquoAdvancesin Difference Equations vol 2012 article 38 2012
[14] R George K P Reshma and R Rajagopalan ldquoA generalisedfixed point theorem of Presic type in cone metric spacesand application to Markov processrdquo Fixed Point Theory andApplications vol 2011 article 85 2011
[15] S Shukla R Sen and S Radenovic ldquoSet-valued Presic typecontraction inmetric spacesrdquoAnalele Stiintifice ale UniversitatiiIn press
[16] S Shukla andR Sen ldquoSet-valued Presic-Reich typemappings inmetric spacesrdquo Revista de la Real Academia de Ciencias ExactasFisicas y Naturales A 2012
[17] S K Malhotra S Shukla and R Sen ldquoA generalization ofBanach contraction principle in ordered cone metric spacesrdquoJournal of Advanced Mathematical Studies vol 5 no 2 pp 59ndash67 2012
[18] Y-Z Chen ldquoA Presic type contractive condition and its applica-tionsrdquo Nonlinear Analysis vol 71 no 12 pp e2012ndashe2017 2009
[19] B E Rhoades ldquoA comparison of various definitions of con-tractive mappingsrdquo Transactions of the American MathematicalSociety vol 226 pp 257ndash290 1977
[20] H Aydi E Karapınar and B Samet ldquoRemarks on some recentfixed point theoremsrdquo Fixed Point Theory and Applications vol2012 article 76 2012
for all 1199091 1199092 119909119896 119909119896+1 isin 119883 where 120572119894 120573119894119895 are non-negative constants such that
119896
sum119894=1
120572119894 + 119896
119896+1
sum119894=1
119896+1
sum119895=1
120573119894119895 lt 1 (18)
Remark 2 Note that for 120573119894119895 = 120573 for all 119894 119895 isin 1 2 119896 119896+1with 119894 = 119895 and 120573119894119894 = 120573119894 for all 119894 isin 1 2 119896 119896+1 the Presic-Hardy-Rogers contraction reduces into the Generalized-Presic contraction With 120573 = 0 the Generalized-Presiccontraction reduces into the Presic-Reich contraction andwith 120572119894 = 0 for all 119894 isin 1 2 119896 120573119894 = 0 for all 119894 isin1 2 119896 119896 + 1 and 120573 = 120574 the Generalized-Presic con-traction reduces into the Presic-Chatterjea contraction With120572119894 = 0 for all 119894 isin 1 2 119896 the Presic-Reich contractionreduces into the Presic-Kannan contraction and with 120573119894 = 0for all 119894 isin 1 2 119896 119896 + 1 the Presic-Reich contractionreduces into the Presic contraction Therefore among allabove definitions the Presic-Hardy-Rogers contraction is themost general contraction
Remark 3 It is easy to see that for 119896 = 1 Presic-Hardy-Rogers contraction reduces into Hardy-Rogers contractionand for 119896 = 1 Generalized-Presic contraction reducesinto Generalized contraction and so forth therefore thecomparison as considered in [19] shows that the abovegeneralization is proper
Now we shall prove some fixed point results for Presic-Hardy-Rogers type contractions in metric spaces
3 Main Results
The following theorem is the fixed point result for Presic-Hardy-Rogers type contractions and the main result of thispaper
Theorem 4 Let (119883 119889) be any complete metric space 119896 apositive integer Let 119891 119883119896 rarr 119883 be a Presic-Hardy-Rogerscontraction then 119891 has a unique fixed point in119883
Proof Let 1199090 isin 119883 be arbitrary Define a sequence 119909119899 in 119883by
119909119899+1 = 119891 (119909119899 119909119899) 119899 ge 0 (19)
If 119909119899 = 119909119899+1 for any 119899 then 119909119899 is a fixed point of 119891 Thereforewe assume 119909119899 = 119909119899+1 for all 119899
We shall show that this sequence is a Cauchy sequence in119883
where119860 119861 119862 119864 and 119865 are the coefficients of 119889119899 119863119899119899 119863119899119899+1119863119899+1119899 and119863119899+1119899+1 respectively in the above inequality
(1 minus 119862 minus 119865) 119889119899+1 le (119860 + 119861 + 119862) 119889119899 (26)
Again as 119889119899+1 = 119889(119909119899 119909119899+1) = 119889(119909119899+1 119909119899) interchanging therole of 119909119899 and 119909119899+1 and repeating above process we obtain
(1 minus 119864 minus 119861) 119889119899+1 le (119860 + 119865 + 119864) 119889119899 (27)
It follows from (26) and (27) that(2 minus 119861 minus 119862 minus 119864 minus 119865) 119889119899+1 le (2119860 + 119861 + 119862 + 119864 + 119865) 119889119899
as 0 le 120582 lt 1 it follows from the above inequalitythat lim119899rarrinfin119889(119909119899 119909119898) = 0 Therefore 119909119899 is a Cauchysequence By completeness of119883 there exists 119906 isin 119883 such thatlim119899rarrinfin119909119899 = 119906
We shall show that 119906 is the fixed point of 119891 Note that
119889 (119906 119891 (119906 119906))
le 119889 (119906 119909119899+1) + 119889 (119909119899+1 119891 (119906 119906))
1 minus 119862 minus 119865119889 (119909119899 119906) +
1 + 119861 + 119864
1 minus 119862 minus 119865119889 (119909119899+1 119906)
(34)
Using the fact that lim119899rarrinfin119909119899 = 119906 it follows from the aboveinequality that
119889 (119906 119891 (119906 119906)) = 0 that is 119891 (119906 119906) = 119906 (35)
Thus 119906 is a fixed point of 119891 For uniqueness let V be anotherfixed point of 119891 that is 119891(V V) = V Again using a similarprocess as used in the calculation of 119889119899+1 we obtain
+ 119862119889 (119906 119891 (V V)) + 119864119889 (V 119891 (119906 119906))
+ 119865119889 (V 119891 (V V))
= (119860 + 119862 + 119864) 119889 (119906 V)
(36)
as 119860+119861 +119862 + 119864 + 119865 lt 1 we obtain 119889(119906 V) = 0 that is 119906 = VThus fixed point is unique
Remark 5 For 119896 = 1 in the above theorem we obtain theresult of Hardy and Rogers [6] For 120573119894119895 = 0 for all 119894 119895 isin1 2 119896 119896 + 1 we obtain the fixed point result of PresicTherefore above theorem is a generalization of the results ofHardy and Rogers and Presic
With Remark 2 the following corollaries are obtained
Corollary 6 Let (119883 119889) be any complete metric space 119896 apositive integer and 119891 119883119896 rarr 119883 a Generalized Presiccontraction Then 119891 has a unique fixed point in119883
For 119896 = 1 in above corollary we obtain the fixed pointresult of Ciric [5]
Corollary 7 Let (119883 119889) be any complete metric space 119896 apositive integer and 119891 119883119896 rarr 119883 a Presic-Reich contractionThen 119891 has a unique fixed point in119883
For 119896 = 1 in the above corollary we obtain the fixed pointresult of Reich [3]
Corollary 8 Let (119883 119889) be any complete metric space 119896 apositive integer and 119891 119883119896 rarr 119883 a Presic-Kannancontraction Then 119891 has a unique fixed point in119883
For 119896 = 1 in above the corollary we obtain the fixed pointresult of Kannan [2]
Corollary 9 Let (119883 119889) be any complete metric space 119896 apositive integer and 119891 119883119896 rarr 119883 a Presic-Chatterjeacontraction Then 119891 has a unique fixed point in119883
For 119896 = 1 in above corollary we obtain the fixed pointresult of Chatterjea [4]
The following are some examples which illustrate thecases when known results are not applicable while our newresults can be used to conclude the existence of fixed point ofmapping
6 Journal of Mathematics
Example 10 Let119883 = [0 1]with usualmetric For 119896 = 2define119891 1198832 rarr 119883 by
119891 (119909 119910) =
1
5 if 119909 = 119910 = 1
119909 + 119910
5 otherwise
(37)
Then
(i) 119891 is a Presic-Reich contraction with 1205721 = 1205722 =
15 1205731 = 1205732 = 1205733 = 111(ii) 119891 is not a Presic contraction(iii) 119891 is not a Presic-Kannan contraction
Proof (i) Note that for 1199091 1199092 1199093 isin [0 1) with 1199091 le 1199092 le 1199093
Therefore conditions (11) and (12) are satisfied for 1205721 = 1205722 =15 and 1205731 1205732 1205733 with 1205731 + 1205732 + 1205733 isin [0 310)
If any one of 1199091 1199092 1199093 is 1 then proof is similar If any twoof 1199091 1199092 1199093 are 1 for example if 1199091 isin [0 1) and 1199092 = 1199093 = 1then
As 1199091 isin [0 1) so conditions (11) and (12) are satisfied for 1205731 =1205732 = 1205733 = 111 and 1205721 1205722 with 1205721 + 1205722 isin [0 511)
Similarly in all possible cases conditions (11) and (12)are satisfied with 1205721 = 1205722 = 15 1205731 = 1205732 = 1205733 =
111 Therefore 119891 is a Presic-Reich contraction All otherconditions of Corollary 7 are satisfied and 0 is the uniquefixed point of 119891
(ii) Note that for 1199091 = 910 and 1199092 = 1199093 = 1
Therefore we cannot find nonnegative constants 1205721 1205722 suchthat condition (7) is satisfied with 1205721 + 1205722 lt 1 So 119891 is not aPresic contraction
Therefore we cannot find nonnegative constant 120573 such thatconditions (9) and (10) are satisfied So 119891 is not a Presic-Kannan contraction
Remark 11 In the above example we cannot apply the resultof Presic [7 8] and Pacurar [11] to conclude the existence offixed point of 119891 But Corollary 7 is applicable which insuresthe existence of unique fixed point of 119891
Example 12 Let 119883 = [0 1] with usual metric For 119896 = 2define 119891 1198832 rarr 119883 by
119891 (119909 119910) =
4
15 if 119909 = 119910 = 1
0 otherwise(42)
Then
(i) 119891 is a Presic-Chatterjea contraction with 120574 isin
[113 112)
(ii) 119891 is not a Presic contraction
(iii) 119891 is not a Presic-Kannan contraction
Proof (i)Note that if1199091 1199092 1199093 isin [0 1) or any one of1199091 1199092 1199093is 1 then conditions (17) and (18) are satisfied trivially
Journal of Mathematics 7
If any two of 1199091 1199092 1199093 are 1 for example if 1199091 isin [0 1)1199092 = 1199093 = 1 then
Therefore we cannot find nonnegative constants 1205721 1205722 suchthat condition (7) is satisfied with 1205721 + 1205722 lt 1 So 119891 is not aPresic contraction
(iii) For 1199091 = 0 1199092 = 1199093 = 1 we have
Therefore we cannot find nonnegative constant 120573 such thatconditions (9) and (10) are satisfied So 119891 is not a Presic-Kannan contraction
Remark 13 In the above example we cannot apply the resultof Presic [7 8] and Pacurar [11] to conclude the existence offixed point of 119891 But Corollary 9 is applicable which insuresthe existence of unique fixed point of 119891
The following theorem is a consequence of Theorem 4and the recent result of Aydi et al [20]
Theorem 14 Let (119883 119889) be any complete metric space and 119896 apositive integer Let 119891 119883119896 rarr 119883 and 119879 119883 rarr 119883 be twomappings such that the following condition holds
Then (119883 120588) is a complete metric space (see [20]) Note thatcondition (46) reduces to the condition (17) that is mapping119891 reduces to Presic-Hardy-Rogers contractionwith respect tometric 120588 So the rest of the proof followedTheorem 4
Acknowledgment
The first author is thankful to Mr Rajpal Tomar for his helpin typing the manuscript
References
[1] R Kannan ldquoSome results on fixed pointsrdquo Bulletin of theCalcutta Mathematical Society vol 60 pp 71ndash76 1968
[2] R Kannan ldquoSome results on fixed points IIrdquo The AmericanMathematical Monthly vol 76 pp 405ndash408 1969
[4] S K Chatterjea ldquoFixed-point theoremsrdquo Comptes Rendus delrsquoAcademie Bulgare des Sciences vol 25 pp 727ndash730 1972
[5] L B Ciric ldquoGeneralized contractions and fixed-point theo-remsrdquo Publications de lrsquoInstitut Mathematique vol 12 no 26pp 19ndash26 1971
8 Journal of Mathematics
[6] G E Hardy and T D Rogers ldquoA generalization of a fixed pointtheorem of Reichrdquo Canadian Mathematical Bulletin vol 16 pp201ndash206 1973
[7] S B Presic ldquoSur une classe drsquoinequations aux differencesfinies et sur la convergence de certaines suitesrdquo Publications delrsquoInstitut Mathematique vol 5 no 19 pp 75ndash78 1965
[8] S B Presic ldquoSur la convergence des suitesrdquo Comptes Rendus delrsquoAcademie des Sciences vol 260 pp 3828ndash3830 1965
[9] L B Ciric and S B Presic ldquoOn Presic type generalization ofthe Banach contraction mapping principlerdquo Acta MathematicaUniversitatis Comenianae vol 76 no 2 pp 143ndash147 2007
[10] M Pacurar ldquoA multi-step iterative method for approximatingcommon fixed points of Presic-Rus type operators on metricspacesrdquo Studia Universitatis Babes-Bolyai Mathematica vol 55no 1 p 149 2010
[11] M Pacurar ldquoApproximating common fixed points of Presic-Kannan type operators by a multi-step iterative methodrdquoAnalele Stiintifice ale Universitatii Ovidius Constanta SeriaMatematica vol 17 no 1 pp 153ndash168 2009
[12] M Pacurar ldquoCommon fixed points for almost Presic typeoperatorsrdquoCarpathian Journal of Mathematics vol 28 no 1 pp117ndash126 2012
[13] M S KhanM Berzig and B Samet ldquoSome convergence resultsfor iterative sequences of Presic type and applicationsrdquoAdvancesin Difference Equations vol 2012 article 38 2012
[14] R George K P Reshma and R Rajagopalan ldquoA generalisedfixed point theorem of Presic type in cone metric spacesand application to Markov processrdquo Fixed Point Theory andApplications vol 2011 article 85 2011
[15] S Shukla R Sen and S Radenovic ldquoSet-valued Presic typecontraction inmetric spacesrdquoAnalele Stiintifice ale UniversitatiiIn press
[16] S Shukla andR Sen ldquoSet-valued Presic-Reich typemappings inmetric spacesrdquo Revista de la Real Academia de Ciencias ExactasFisicas y Naturales A 2012
[17] S K Malhotra S Shukla and R Sen ldquoA generalization ofBanach contraction principle in ordered cone metric spacesrdquoJournal of Advanced Mathematical Studies vol 5 no 2 pp 59ndash67 2012
[18] Y-Z Chen ldquoA Presic type contractive condition and its applica-tionsrdquo Nonlinear Analysis vol 71 no 12 pp e2012ndashe2017 2009
[19] B E Rhoades ldquoA comparison of various definitions of con-tractive mappingsrdquo Transactions of the American MathematicalSociety vol 226 pp 257ndash290 1977
[20] H Aydi E Karapınar and B Samet ldquoRemarks on some recentfixed point theoremsrdquo Fixed Point Theory and Applications vol2012 article 76 2012
where119860 119861 119862 119864 and 119865 are the coefficients of 119889119899 119863119899119899 119863119899119899+1119863119899+1119899 and119863119899+1119899+1 respectively in the above inequality
(1 minus 119862 minus 119865) 119889119899+1 le (119860 + 119861 + 119862) 119889119899 (26)
Again as 119889119899+1 = 119889(119909119899 119909119899+1) = 119889(119909119899+1 119909119899) interchanging therole of 119909119899 and 119909119899+1 and repeating above process we obtain
(1 minus 119864 minus 119861) 119889119899+1 le (119860 + 119865 + 119864) 119889119899 (27)
It follows from (26) and (27) that(2 minus 119861 minus 119862 minus 119864 minus 119865) 119889119899+1 le (2119860 + 119861 + 119862 + 119864 + 119865) 119889119899
as 0 le 120582 lt 1 it follows from the above inequalitythat lim119899rarrinfin119889(119909119899 119909119898) = 0 Therefore 119909119899 is a Cauchysequence By completeness of119883 there exists 119906 isin 119883 such thatlim119899rarrinfin119909119899 = 119906
We shall show that 119906 is the fixed point of 119891 Note that
119889 (119906 119891 (119906 119906))
le 119889 (119906 119909119899+1) + 119889 (119909119899+1 119891 (119906 119906))
1 minus 119862 minus 119865119889 (119909119899 119906) +
1 + 119861 + 119864
1 minus 119862 minus 119865119889 (119909119899+1 119906)
(34)
Using the fact that lim119899rarrinfin119909119899 = 119906 it follows from the aboveinequality that
119889 (119906 119891 (119906 119906)) = 0 that is 119891 (119906 119906) = 119906 (35)
Thus 119906 is a fixed point of 119891 For uniqueness let V be anotherfixed point of 119891 that is 119891(V V) = V Again using a similarprocess as used in the calculation of 119889119899+1 we obtain
+ 119862119889 (119906 119891 (V V)) + 119864119889 (V 119891 (119906 119906))
+ 119865119889 (V 119891 (V V))
= (119860 + 119862 + 119864) 119889 (119906 V)
(36)
as 119860+119861 +119862 + 119864 + 119865 lt 1 we obtain 119889(119906 V) = 0 that is 119906 = VThus fixed point is unique
Remark 5 For 119896 = 1 in the above theorem we obtain theresult of Hardy and Rogers [6] For 120573119894119895 = 0 for all 119894 119895 isin1 2 119896 119896 + 1 we obtain the fixed point result of PresicTherefore above theorem is a generalization of the results ofHardy and Rogers and Presic
With Remark 2 the following corollaries are obtained
Corollary 6 Let (119883 119889) be any complete metric space 119896 apositive integer and 119891 119883119896 rarr 119883 a Generalized Presiccontraction Then 119891 has a unique fixed point in119883
For 119896 = 1 in above corollary we obtain the fixed pointresult of Ciric [5]
Corollary 7 Let (119883 119889) be any complete metric space 119896 apositive integer and 119891 119883119896 rarr 119883 a Presic-Reich contractionThen 119891 has a unique fixed point in119883
For 119896 = 1 in the above corollary we obtain the fixed pointresult of Reich [3]
Corollary 8 Let (119883 119889) be any complete metric space 119896 apositive integer and 119891 119883119896 rarr 119883 a Presic-Kannancontraction Then 119891 has a unique fixed point in119883
For 119896 = 1 in above the corollary we obtain the fixed pointresult of Kannan [2]
Corollary 9 Let (119883 119889) be any complete metric space 119896 apositive integer and 119891 119883119896 rarr 119883 a Presic-Chatterjeacontraction Then 119891 has a unique fixed point in119883
For 119896 = 1 in above corollary we obtain the fixed pointresult of Chatterjea [4]
The following are some examples which illustrate thecases when known results are not applicable while our newresults can be used to conclude the existence of fixed point ofmapping
6 Journal of Mathematics
Example 10 Let119883 = [0 1]with usualmetric For 119896 = 2define119891 1198832 rarr 119883 by
119891 (119909 119910) =
1
5 if 119909 = 119910 = 1
119909 + 119910
5 otherwise
(37)
Then
(i) 119891 is a Presic-Reich contraction with 1205721 = 1205722 =
15 1205731 = 1205732 = 1205733 = 111(ii) 119891 is not a Presic contraction(iii) 119891 is not a Presic-Kannan contraction
Proof (i) Note that for 1199091 1199092 1199093 isin [0 1) with 1199091 le 1199092 le 1199093
Therefore conditions (11) and (12) are satisfied for 1205721 = 1205722 =15 and 1205731 1205732 1205733 with 1205731 + 1205732 + 1205733 isin [0 310)
If any one of 1199091 1199092 1199093 is 1 then proof is similar If any twoof 1199091 1199092 1199093 are 1 for example if 1199091 isin [0 1) and 1199092 = 1199093 = 1then
As 1199091 isin [0 1) so conditions (11) and (12) are satisfied for 1205731 =1205732 = 1205733 = 111 and 1205721 1205722 with 1205721 + 1205722 isin [0 511)
Similarly in all possible cases conditions (11) and (12)are satisfied with 1205721 = 1205722 = 15 1205731 = 1205732 = 1205733 =
111 Therefore 119891 is a Presic-Reich contraction All otherconditions of Corollary 7 are satisfied and 0 is the uniquefixed point of 119891
(ii) Note that for 1199091 = 910 and 1199092 = 1199093 = 1
Therefore we cannot find nonnegative constants 1205721 1205722 suchthat condition (7) is satisfied with 1205721 + 1205722 lt 1 So 119891 is not aPresic contraction
Therefore we cannot find nonnegative constant 120573 such thatconditions (9) and (10) are satisfied So 119891 is not a Presic-Kannan contraction
Remark 11 In the above example we cannot apply the resultof Presic [7 8] and Pacurar [11] to conclude the existence offixed point of 119891 But Corollary 7 is applicable which insuresthe existence of unique fixed point of 119891
Example 12 Let 119883 = [0 1] with usual metric For 119896 = 2define 119891 1198832 rarr 119883 by
119891 (119909 119910) =
4
15 if 119909 = 119910 = 1
0 otherwise(42)
Then
(i) 119891 is a Presic-Chatterjea contraction with 120574 isin
[113 112)
(ii) 119891 is not a Presic contraction
(iii) 119891 is not a Presic-Kannan contraction
Proof (i)Note that if1199091 1199092 1199093 isin [0 1) or any one of1199091 1199092 1199093is 1 then conditions (17) and (18) are satisfied trivially
Journal of Mathematics 7
If any two of 1199091 1199092 1199093 are 1 for example if 1199091 isin [0 1)1199092 = 1199093 = 1 then
Therefore we cannot find nonnegative constants 1205721 1205722 suchthat condition (7) is satisfied with 1205721 + 1205722 lt 1 So 119891 is not aPresic contraction
(iii) For 1199091 = 0 1199092 = 1199093 = 1 we have
Therefore we cannot find nonnegative constant 120573 such thatconditions (9) and (10) are satisfied So 119891 is not a Presic-Kannan contraction
Remark 13 In the above example we cannot apply the resultof Presic [7 8] and Pacurar [11] to conclude the existence offixed point of 119891 But Corollary 9 is applicable which insuresthe existence of unique fixed point of 119891
The following theorem is a consequence of Theorem 4and the recent result of Aydi et al [20]
Theorem 14 Let (119883 119889) be any complete metric space and 119896 apositive integer Let 119891 119883119896 rarr 119883 and 119879 119883 rarr 119883 be twomappings such that the following condition holds
Then (119883 120588) is a complete metric space (see [20]) Note thatcondition (46) reduces to the condition (17) that is mapping119891 reduces to Presic-Hardy-Rogers contractionwith respect tometric 120588 So the rest of the proof followedTheorem 4
Acknowledgment
The first author is thankful to Mr Rajpal Tomar for his helpin typing the manuscript
References
[1] R Kannan ldquoSome results on fixed pointsrdquo Bulletin of theCalcutta Mathematical Society vol 60 pp 71ndash76 1968
[2] R Kannan ldquoSome results on fixed points IIrdquo The AmericanMathematical Monthly vol 76 pp 405ndash408 1969
[4] S K Chatterjea ldquoFixed-point theoremsrdquo Comptes Rendus delrsquoAcademie Bulgare des Sciences vol 25 pp 727ndash730 1972
[5] L B Ciric ldquoGeneralized contractions and fixed-point theo-remsrdquo Publications de lrsquoInstitut Mathematique vol 12 no 26pp 19ndash26 1971
8 Journal of Mathematics
[6] G E Hardy and T D Rogers ldquoA generalization of a fixed pointtheorem of Reichrdquo Canadian Mathematical Bulletin vol 16 pp201ndash206 1973
[7] S B Presic ldquoSur une classe drsquoinequations aux differencesfinies et sur la convergence de certaines suitesrdquo Publications delrsquoInstitut Mathematique vol 5 no 19 pp 75ndash78 1965
[8] S B Presic ldquoSur la convergence des suitesrdquo Comptes Rendus delrsquoAcademie des Sciences vol 260 pp 3828ndash3830 1965
[9] L B Ciric and S B Presic ldquoOn Presic type generalization ofthe Banach contraction mapping principlerdquo Acta MathematicaUniversitatis Comenianae vol 76 no 2 pp 143ndash147 2007
[10] M Pacurar ldquoA multi-step iterative method for approximatingcommon fixed points of Presic-Rus type operators on metricspacesrdquo Studia Universitatis Babes-Bolyai Mathematica vol 55no 1 p 149 2010
[11] M Pacurar ldquoApproximating common fixed points of Presic-Kannan type operators by a multi-step iterative methodrdquoAnalele Stiintifice ale Universitatii Ovidius Constanta SeriaMatematica vol 17 no 1 pp 153ndash168 2009
[12] M Pacurar ldquoCommon fixed points for almost Presic typeoperatorsrdquoCarpathian Journal of Mathematics vol 28 no 1 pp117ndash126 2012
[13] M S KhanM Berzig and B Samet ldquoSome convergence resultsfor iterative sequences of Presic type and applicationsrdquoAdvancesin Difference Equations vol 2012 article 38 2012
[14] R George K P Reshma and R Rajagopalan ldquoA generalisedfixed point theorem of Presic type in cone metric spacesand application to Markov processrdquo Fixed Point Theory andApplications vol 2011 article 85 2011
[15] S Shukla R Sen and S Radenovic ldquoSet-valued Presic typecontraction inmetric spacesrdquoAnalele Stiintifice ale UniversitatiiIn press
[16] S Shukla andR Sen ldquoSet-valued Presic-Reich typemappings inmetric spacesrdquo Revista de la Real Academia de Ciencias ExactasFisicas y Naturales A 2012
[17] S K Malhotra S Shukla and R Sen ldquoA generalization ofBanach contraction principle in ordered cone metric spacesrdquoJournal of Advanced Mathematical Studies vol 5 no 2 pp 59ndash67 2012
[18] Y-Z Chen ldquoA Presic type contractive condition and its applica-tionsrdquo Nonlinear Analysis vol 71 no 12 pp e2012ndashe2017 2009
[19] B E Rhoades ldquoA comparison of various definitions of con-tractive mappingsrdquo Transactions of the American MathematicalSociety vol 226 pp 257ndash290 1977
[20] H Aydi E Karapınar and B Samet ldquoRemarks on some recentfixed point theoremsrdquo Fixed Point Theory and Applications vol2012 article 76 2012
as 0 le 120582 lt 1 it follows from the above inequalitythat lim119899rarrinfin119889(119909119899 119909119898) = 0 Therefore 119909119899 is a Cauchysequence By completeness of119883 there exists 119906 isin 119883 such thatlim119899rarrinfin119909119899 = 119906
We shall show that 119906 is the fixed point of 119891 Note that
119889 (119906 119891 (119906 119906))
le 119889 (119906 119909119899+1) + 119889 (119909119899+1 119891 (119906 119906))
1 minus 119862 minus 119865119889 (119909119899 119906) +
1 + 119861 + 119864
1 minus 119862 minus 119865119889 (119909119899+1 119906)
(34)
Using the fact that lim119899rarrinfin119909119899 = 119906 it follows from the aboveinequality that
119889 (119906 119891 (119906 119906)) = 0 that is 119891 (119906 119906) = 119906 (35)
Thus 119906 is a fixed point of 119891 For uniqueness let V be anotherfixed point of 119891 that is 119891(V V) = V Again using a similarprocess as used in the calculation of 119889119899+1 we obtain
+ 119862119889 (119906 119891 (V V)) + 119864119889 (V 119891 (119906 119906))
+ 119865119889 (V 119891 (V V))
= (119860 + 119862 + 119864) 119889 (119906 V)
(36)
as 119860+119861 +119862 + 119864 + 119865 lt 1 we obtain 119889(119906 V) = 0 that is 119906 = VThus fixed point is unique
Remark 5 For 119896 = 1 in the above theorem we obtain theresult of Hardy and Rogers [6] For 120573119894119895 = 0 for all 119894 119895 isin1 2 119896 119896 + 1 we obtain the fixed point result of PresicTherefore above theorem is a generalization of the results ofHardy and Rogers and Presic
With Remark 2 the following corollaries are obtained
Corollary 6 Let (119883 119889) be any complete metric space 119896 apositive integer and 119891 119883119896 rarr 119883 a Generalized Presiccontraction Then 119891 has a unique fixed point in119883
For 119896 = 1 in above corollary we obtain the fixed pointresult of Ciric [5]
Corollary 7 Let (119883 119889) be any complete metric space 119896 apositive integer and 119891 119883119896 rarr 119883 a Presic-Reich contractionThen 119891 has a unique fixed point in119883
For 119896 = 1 in the above corollary we obtain the fixed pointresult of Reich [3]
Corollary 8 Let (119883 119889) be any complete metric space 119896 apositive integer and 119891 119883119896 rarr 119883 a Presic-Kannancontraction Then 119891 has a unique fixed point in119883
For 119896 = 1 in above the corollary we obtain the fixed pointresult of Kannan [2]
Corollary 9 Let (119883 119889) be any complete metric space 119896 apositive integer and 119891 119883119896 rarr 119883 a Presic-Chatterjeacontraction Then 119891 has a unique fixed point in119883
For 119896 = 1 in above corollary we obtain the fixed pointresult of Chatterjea [4]
The following are some examples which illustrate thecases when known results are not applicable while our newresults can be used to conclude the existence of fixed point ofmapping
6 Journal of Mathematics
Example 10 Let119883 = [0 1]with usualmetric For 119896 = 2define119891 1198832 rarr 119883 by
119891 (119909 119910) =
1
5 if 119909 = 119910 = 1
119909 + 119910
5 otherwise
(37)
Then
(i) 119891 is a Presic-Reich contraction with 1205721 = 1205722 =
15 1205731 = 1205732 = 1205733 = 111(ii) 119891 is not a Presic contraction(iii) 119891 is not a Presic-Kannan contraction
Proof (i) Note that for 1199091 1199092 1199093 isin [0 1) with 1199091 le 1199092 le 1199093
Therefore conditions (11) and (12) are satisfied for 1205721 = 1205722 =15 and 1205731 1205732 1205733 with 1205731 + 1205732 + 1205733 isin [0 310)
If any one of 1199091 1199092 1199093 is 1 then proof is similar If any twoof 1199091 1199092 1199093 are 1 for example if 1199091 isin [0 1) and 1199092 = 1199093 = 1then
As 1199091 isin [0 1) so conditions (11) and (12) are satisfied for 1205731 =1205732 = 1205733 = 111 and 1205721 1205722 with 1205721 + 1205722 isin [0 511)
Similarly in all possible cases conditions (11) and (12)are satisfied with 1205721 = 1205722 = 15 1205731 = 1205732 = 1205733 =
111 Therefore 119891 is a Presic-Reich contraction All otherconditions of Corollary 7 are satisfied and 0 is the uniquefixed point of 119891
(ii) Note that for 1199091 = 910 and 1199092 = 1199093 = 1
Therefore we cannot find nonnegative constants 1205721 1205722 suchthat condition (7) is satisfied with 1205721 + 1205722 lt 1 So 119891 is not aPresic contraction
Therefore we cannot find nonnegative constant 120573 such thatconditions (9) and (10) are satisfied So 119891 is not a Presic-Kannan contraction
Remark 11 In the above example we cannot apply the resultof Presic [7 8] and Pacurar [11] to conclude the existence offixed point of 119891 But Corollary 7 is applicable which insuresthe existence of unique fixed point of 119891
Example 12 Let 119883 = [0 1] with usual metric For 119896 = 2define 119891 1198832 rarr 119883 by
119891 (119909 119910) =
4
15 if 119909 = 119910 = 1
0 otherwise(42)
Then
(i) 119891 is a Presic-Chatterjea contraction with 120574 isin
[113 112)
(ii) 119891 is not a Presic contraction
(iii) 119891 is not a Presic-Kannan contraction
Proof (i)Note that if1199091 1199092 1199093 isin [0 1) or any one of1199091 1199092 1199093is 1 then conditions (17) and (18) are satisfied trivially
Journal of Mathematics 7
If any two of 1199091 1199092 1199093 are 1 for example if 1199091 isin [0 1)1199092 = 1199093 = 1 then
Therefore we cannot find nonnegative constants 1205721 1205722 suchthat condition (7) is satisfied with 1205721 + 1205722 lt 1 So 119891 is not aPresic contraction
(iii) For 1199091 = 0 1199092 = 1199093 = 1 we have
Therefore we cannot find nonnegative constant 120573 such thatconditions (9) and (10) are satisfied So 119891 is not a Presic-Kannan contraction
Remark 13 In the above example we cannot apply the resultof Presic [7 8] and Pacurar [11] to conclude the existence offixed point of 119891 But Corollary 9 is applicable which insuresthe existence of unique fixed point of 119891
The following theorem is a consequence of Theorem 4and the recent result of Aydi et al [20]
Theorem 14 Let (119883 119889) be any complete metric space and 119896 apositive integer Let 119891 119883119896 rarr 119883 and 119879 119883 rarr 119883 be twomappings such that the following condition holds
Then (119883 120588) is a complete metric space (see [20]) Note thatcondition (46) reduces to the condition (17) that is mapping119891 reduces to Presic-Hardy-Rogers contractionwith respect tometric 120588 So the rest of the proof followedTheorem 4
Acknowledgment
The first author is thankful to Mr Rajpal Tomar for his helpin typing the manuscript
References
[1] R Kannan ldquoSome results on fixed pointsrdquo Bulletin of theCalcutta Mathematical Society vol 60 pp 71ndash76 1968
[2] R Kannan ldquoSome results on fixed points IIrdquo The AmericanMathematical Monthly vol 76 pp 405ndash408 1969
[4] S K Chatterjea ldquoFixed-point theoremsrdquo Comptes Rendus delrsquoAcademie Bulgare des Sciences vol 25 pp 727ndash730 1972
[5] L B Ciric ldquoGeneralized contractions and fixed-point theo-remsrdquo Publications de lrsquoInstitut Mathematique vol 12 no 26pp 19ndash26 1971
8 Journal of Mathematics
[6] G E Hardy and T D Rogers ldquoA generalization of a fixed pointtheorem of Reichrdquo Canadian Mathematical Bulletin vol 16 pp201ndash206 1973
[7] S B Presic ldquoSur une classe drsquoinequations aux differencesfinies et sur la convergence de certaines suitesrdquo Publications delrsquoInstitut Mathematique vol 5 no 19 pp 75ndash78 1965
[8] S B Presic ldquoSur la convergence des suitesrdquo Comptes Rendus delrsquoAcademie des Sciences vol 260 pp 3828ndash3830 1965
[9] L B Ciric and S B Presic ldquoOn Presic type generalization ofthe Banach contraction mapping principlerdquo Acta MathematicaUniversitatis Comenianae vol 76 no 2 pp 143ndash147 2007
[10] M Pacurar ldquoA multi-step iterative method for approximatingcommon fixed points of Presic-Rus type operators on metricspacesrdquo Studia Universitatis Babes-Bolyai Mathematica vol 55no 1 p 149 2010
[11] M Pacurar ldquoApproximating common fixed points of Presic-Kannan type operators by a multi-step iterative methodrdquoAnalele Stiintifice ale Universitatii Ovidius Constanta SeriaMatematica vol 17 no 1 pp 153ndash168 2009
[12] M Pacurar ldquoCommon fixed points for almost Presic typeoperatorsrdquoCarpathian Journal of Mathematics vol 28 no 1 pp117ndash126 2012
[13] M S KhanM Berzig and B Samet ldquoSome convergence resultsfor iterative sequences of Presic type and applicationsrdquoAdvancesin Difference Equations vol 2012 article 38 2012
[14] R George K P Reshma and R Rajagopalan ldquoA generalisedfixed point theorem of Presic type in cone metric spacesand application to Markov processrdquo Fixed Point Theory andApplications vol 2011 article 85 2011
[15] S Shukla R Sen and S Radenovic ldquoSet-valued Presic typecontraction inmetric spacesrdquoAnalele Stiintifice ale UniversitatiiIn press
[16] S Shukla andR Sen ldquoSet-valued Presic-Reich typemappings inmetric spacesrdquo Revista de la Real Academia de Ciencias ExactasFisicas y Naturales A 2012
[17] S K Malhotra S Shukla and R Sen ldquoA generalization ofBanach contraction principle in ordered cone metric spacesrdquoJournal of Advanced Mathematical Studies vol 5 no 2 pp 59ndash67 2012
[18] Y-Z Chen ldquoA Presic type contractive condition and its applica-tionsrdquo Nonlinear Analysis vol 71 no 12 pp e2012ndashe2017 2009
[19] B E Rhoades ldquoA comparison of various definitions of con-tractive mappingsrdquo Transactions of the American MathematicalSociety vol 226 pp 257ndash290 1977
[20] H Aydi E Karapınar and B Samet ldquoRemarks on some recentfixed point theoremsrdquo Fixed Point Theory and Applications vol2012 article 76 2012
Therefore conditions (11) and (12) are satisfied for 1205721 = 1205722 =15 and 1205731 1205732 1205733 with 1205731 + 1205732 + 1205733 isin [0 310)
If any one of 1199091 1199092 1199093 is 1 then proof is similar If any twoof 1199091 1199092 1199093 are 1 for example if 1199091 isin [0 1) and 1199092 = 1199093 = 1then
As 1199091 isin [0 1) so conditions (11) and (12) are satisfied for 1205731 =1205732 = 1205733 = 111 and 1205721 1205722 with 1205721 + 1205722 isin [0 511)
Similarly in all possible cases conditions (11) and (12)are satisfied with 1205721 = 1205722 = 15 1205731 = 1205732 = 1205733 =
111 Therefore 119891 is a Presic-Reich contraction All otherconditions of Corollary 7 are satisfied and 0 is the uniquefixed point of 119891
(ii) Note that for 1199091 = 910 and 1199092 = 1199093 = 1
Therefore we cannot find nonnegative constants 1205721 1205722 suchthat condition (7) is satisfied with 1205721 + 1205722 lt 1 So 119891 is not aPresic contraction
Therefore we cannot find nonnegative constant 120573 such thatconditions (9) and (10) are satisfied So 119891 is not a Presic-Kannan contraction
Remark 11 In the above example we cannot apply the resultof Presic [7 8] and Pacurar [11] to conclude the existence offixed point of 119891 But Corollary 7 is applicable which insuresthe existence of unique fixed point of 119891
Example 12 Let 119883 = [0 1] with usual metric For 119896 = 2define 119891 1198832 rarr 119883 by
119891 (119909 119910) =
4
15 if 119909 = 119910 = 1
0 otherwise(42)
Then
(i) 119891 is a Presic-Chatterjea contraction with 120574 isin
[113 112)
(ii) 119891 is not a Presic contraction
(iii) 119891 is not a Presic-Kannan contraction
Proof (i)Note that if1199091 1199092 1199093 isin [0 1) or any one of1199091 1199092 1199093is 1 then conditions (17) and (18) are satisfied trivially
Journal of Mathematics 7
If any two of 1199091 1199092 1199093 are 1 for example if 1199091 isin [0 1)1199092 = 1199093 = 1 then
Therefore we cannot find nonnegative constants 1205721 1205722 suchthat condition (7) is satisfied with 1205721 + 1205722 lt 1 So 119891 is not aPresic contraction
(iii) For 1199091 = 0 1199092 = 1199093 = 1 we have
Therefore we cannot find nonnegative constant 120573 such thatconditions (9) and (10) are satisfied So 119891 is not a Presic-Kannan contraction
Remark 13 In the above example we cannot apply the resultof Presic [7 8] and Pacurar [11] to conclude the existence offixed point of 119891 But Corollary 9 is applicable which insuresthe existence of unique fixed point of 119891
The following theorem is a consequence of Theorem 4and the recent result of Aydi et al [20]
Theorem 14 Let (119883 119889) be any complete metric space and 119896 apositive integer Let 119891 119883119896 rarr 119883 and 119879 119883 rarr 119883 be twomappings such that the following condition holds
Then (119883 120588) is a complete metric space (see [20]) Note thatcondition (46) reduces to the condition (17) that is mapping119891 reduces to Presic-Hardy-Rogers contractionwith respect tometric 120588 So the rest of the proof followedTheorem 4
Acknowledgment
The first author is thankful to Mr Rajpal Tomar for his helpin typing the manuscript
References
[1] R Kannan ldquoSome results on fixed pointsrdquo Bulletin of theCalcutta Mathematical Society vol 60 pp 71ndash76 1968
[2] R Kannan ldquoSome results on fixed points IIrdquo The AmericanMathematical Monthly vol 76 pp 405ndash408 1969
[4] S K Chatterjea ldquoFixed-point theoremsrdquo Comptes Rendus delrsquoAcademie Bulgare des Sciences vol 25 pp 727ndash730 1972
[5] L B Ciric ldquoGeneralized contractions and fixed-point theo-remsrdquo Publications de lrsquoInstitut Mathematique vol 12 no 26pp 19ndash26 1971
8 Journal of Mathematics
[6] G E Hardy and T D Rogers ldquoA generalization of a fixed pointtheorem of Reichrdquo Canadian Mathematical Bulletin vol 16 pp201ndash206 1973
[7] S B Presic ldquoSur une classe drsquoinequations aux differencesfinies et sur la convergence de certaines suitesrdquo Publications delrsquoInstitut Mathematique vol 5 no 19 pp 75ndash78 1965
[8] S B Presic ldquoSur la convergence des suitesrdquo Comptes Rendus delrsquoAcademie des Sciences vol 260 pp 3828ndash3830 1965
[9] L B Ciric and S B Presic ldquoOn Presic type generalization ofthe Banach contraction mapping principlerdquo Acta MathematicaUniversitatis Comenianae vol 76 no 2 pp 143ndash147 2007
[10] M Pacurar ldquoA multi-step iterative method for approximatingcommon fixed points of Presic-Rus type operators on metricspacesrdquo Studia Universitatis Babes-Bolyai Mathematica vol 55no 1 p 149 2010
[11] M Pacurar ldquoApproximating common fixed points of Presic-Kannan type operators by a multi-step iterative methodrdquoAnalele Stiintifice ale Universitatii Ovidius Constanta SeriaMatematica vol 17 no 1 pp 153ndash168 2009
[12] M Pacurar ldquoCommon fixed points for almost Presic typeoperatorsrdquoCarpathian Journal of Mathematics vol 28 no 1 pp117ndash126 2012
[13] M S KhanM Berzig and B Samet ldquoSome convergence resultsfor iterative sequences of Presic type and applicationsrdquoAdvancesin Difference Equations vol 2012 article 38 2012
[14] R George K P Reshma and R Rajagopalan ldquoA generalisedfixed point theorem of Presic type in cone metric spacesand application to Markov processrdquo Fixed Point Theory andApplications vol 2011 article 85 2011
[15] S Shukla R Sen and S Radenovic ldquoSet-valued Presic typecontraction inmetric spacesrdquoAnalele Stiintifice ale UniversitatiiIn press
[16] S Shukla andR Sen ldquoSet-valued Presic-Reich typemappings inmetric spacesrdquo Revista de la Real Academia de Ciencias ExactasFisicas y Naturales A 2012
[17] S K Malhotra S Shukla and R Sen ldquoA generalization ofBanach contraction principle in ordered cone metric spacesrdquoJournal of Advanced Mathematical Studies vol 5 no 2 pp 59ndash67 2012
[18] Y-Z Chen ldquoA Presic type contractive condition and its applica-tionsrdquo Nonlinear Analysis vol 71 no 12 pp e2012ndashe2017 2009
[19] B E Rhoades ldquoA comparison of various definitions of con-tractive mappingsrdquo Transactions of the American MathematicalSociety vol 226 pp 257ndash290 1977
[20] H Aydi E Karapınar and B Samet ldquoRemarks on some recentfixed point theoremsrdquo Fixed Point Theory and Applications vol2012 article 76 2012
Therefore we cannot find nonnegative constants 1205721 1205722 suchthat condition (7) is satisfied with 1205721 + 1205722 lt 1 So 119891 is not aPresic contraction
(iii) For 1199091 = 0 1199092 = 1199093 = 1 we have
Therefore we cannot find nonnegative constant 120573 such thatconditions (9) and (10) are satisfied So 119891 is not a Presic-Kannan contraction
Remark 13 In the above example we cannot apply the resultof Presic [7 8] and Pacurar [11] to conclude the existence offixed point of 119891 But Corollary 9 is applicable which insuresthe existence of unique fixed point of 119891
The following theorem is a consequence of Theorem 4and the recent result of Aydi et al [20]
Theorem 14 Let (119883 119889) be any complete metric space and 119896 apositive integer Let 119891 119883119896 rarr 119883 and 119879 119883 rarr 119883 be twomappings such that the following condition holds
Then (119883 120588) is a complete metric space (see [20]) Note thatcondition (46) reduces to the condition (17) that is mapping119891 reduces to Presic-Hardy-Rogers contractionwith respect tometric 120588 So the rest of the proof followedTheorem 4
Acknowledgment
The first author is thankful to Mr Rajpal Tomar for his helpin typing the manuscript
References
[1] R Kannan ldquoSome results on fixed pointsrdquo Bulletin of theCalcutta Mathematical Society vol 60 pp 71ndash76 1968
[2] R Kannan ldquoSome results on fixed points IIrdquo The AmericanMathematical Monthly vol 76 pp 405ndash408 1969
[4] S K Chatterjea ldquoFixed-point theoremsrdquo Comptes Rendus delrsquoAcademie Bulgare des Sciences vol 25 pp 727ndash730 1972
[5] L B Ciric ldquoGeneralized contractions and fixed-point theo-remsrdquo Publications de lrsquoInstitut Mathematique vol 12 no 26pp 19ndash26 1971
8 Journal of Mathematics
[6] G E Hardy and T D Rogers ldquoA generalization of a fixed pointtheorem of Reichrdquo Canadian Mathematical Bulletin vol 16 pp201ndash206 1973
[7] S B Presic ldquoSur une classe drsquoinequations aux differencesfinies et sur la convergence de certaines suitesrdquo Publications delrsquoInstitut Mathematique vol 5 no 19 pp 75ndash78 1965
[8] S B Presic ldquoSur la convergence des suitesrdquo Comptes Rendus delrsquoAcademie des Sciences vol 260 pp 3828ndash3830 1965
[9] L B Ciric and S B Presic ldquoOn Presic type generalization ofthe Banach contraction mapping principlerdquo Acta MathematicaUniversitatis Comenianae vol 76 no 2 pp 143ndash147 2007
[10] M Pacurar ldquoA multi-step iterative method for approximatingcommon fixed points of Presic-Rus type operators on metricspacesrdquo Studia Universitatis Babes-Bolyai Mathematica vol 55no 1 p 149 2010
[11] M Pacurar ldquoApproximating common fixed points of Presic-Kannan type operators by a multi-step iterative methodrdquoAnalele Stiintifice ale Universitatii Ovidius Constanta SeriaMatematica vol 17 no 1 pp 153ndash168 2009
[12] M Pacurar ldquoCommon fixed points for almost Presic typeoperatorsrdquoCarpathian Journal of Mathematics vol 28 no 1 pp117ndash126 2012
[13] M S KhanM Berzig and B Samet ldquoSome convergence resultsfor iterative sequences of Presic type and applicationsrdquoAdvancesin Difference Equations vol 2012 article 38 2012
[14] R George K P Reshma and R Rajagopalan ldquoA generalisedfixed point theorem of Presic type in cone metric spacesand application to Markov processrdquo Fixed Point Theory andApplications vol 2011 article 85 2011
[15] S Shukla R Sen and S Radenovic ldquoSet-valued Presic typecontraction inmetric spacesrdquoAnalele Stiintifice ale UniversitatiiIn press
[16] S Shukla andR Sen ldquoSet-valued Presic-Reich typemappings inmetric spacesrdquo Revista de la Real Academia de Ciencias ExactasFisicas y Naturales A 2012
[17] S K Malhotra S Shukla and R Sen ldquoA generalization ofBanach contraction principle in ordered cone metric spacesrdquoJournal of Advanced Mathematical Studies vol 5 no 2 pp 59ndash67 2012
[18] Y-Z Chen ldquoA Presic type contractive condition and its applica-tionsrdquo Nonlinear Analysis vol 71 no 12 pp e2012ndashe2017 2009
[19] B E Rhoades ldquoA comparison of various definitions of con-tractive mappingsrdquo Transactions of the American MathematicalSociety vol 226 pp 257ndash290 1977
[20] H Aydi E Karapınar and B Samet ldquoRemarks on some recentfixed point theoremsrdquo Fixed Point Theory and Applications vol2012 article 76 2012
[6] G E Hardy and T D Rogers ldquoA generalization of a fixed pointtheorem of Reichrdquo Canadian Mathematical Bulletin vol 16 pp201ndash206 1973
[7] S B Presic ldquoSur une classe drsquoinequations aux differencesfinies et sur la convergence de certaines suitesrdquo Publications delrsquoInstitut Mathematique vol 5 no 19 pp 75ndash78 1965
[8] S B Presic ldquoSur la convergence des suitesrdquo Comptes Rendus delrsquoAcademie des Sciences vol 260 pp 3828ndash3830 1965
[9] L B Ciric and S B Presic ldquoOn Presic type generalization ofthe Banach contraction mapping principlerdquo Acta MathematicaUniversitatis Comenianae vol 76 no 2 pp 143ndash147 2007
[10] M Pacurar ldquoA multi-step iterative method for approximatingcommon fixed points of Presic-Rus type operators on metricspacesrdquo Studia Universitatis Babes-Bolyai Mathematica vol 55no 1 p 149 2010
[11] M Pacurar ldquoApproximating common fixed points of Presic-Kannan type operators by a multi-step iterative methodrdquoAnalele Stiintifice ale Universitatii Ovidius Constanta SeriaMatematica vol 17 no 1 pp 153ndash168 2009
[12] M Pacurar ldquoCommon fixed points for almost Presic typeoperatorsrdquoCarpathian Journal of Mathematics vol 28 no 1 pp117ndash126 2012
[13] M S KhanM Berzig and B Samet ldquoSome convergence resultsfor iterative sequences of Presic type and applicationsrdquoAdvancesin Difference Equations vol 2012 article 38 2012
[14] R George K P Reshma and R Rajagopalan ldquoA generalisedfixed point theorem of Presic type in cone metric spacesand application to Markov processrdquo Fixed Point Theory andApplications vol 2011 article 85 2011
[15] S Shukla R Sen and S Radenovic ldquoSet-valued Presic typecontraction inmetric spacesrdquoAnalele Stiintifice ale UniversitatiiIn press
[16] S Shukla andR Sen ldquoSet-valued Presic-Reich typemappings inmetric spacesrdquo Revista de la Real Academia de Ciencias ExactasFisicas y Naturales A 2012
[17] S K Malhotra S Shukla and R Sen ldquoA generalization ofBanach contraction principle in ordered cone metric spacesrdquoJournal of Advanced Mathematical Studies vol 5 no 2 pp 59ndash67 2012
[18] Y-Z Chen ldquoA Presic type contractive condition and its applica-tionsrdquo Nonlinear Analysis vol 71 no 12 pp e2012ndashe2017 2009
[19] B E Rhoades ldquoA comparison of various definitions of con-tractive mappingsrdquo Transactions of the American MathematicalSociety vol 226 pp 257ndash290 1977
[20] H Aydi E Karapınar and B Samet ldquoRemarks on some recentfixed point theoremsrdquo Fixed Point Theory and Applications vol2012 article 76 2012
