Research ArticleTwo New Types of Fixed Point Theorems inComplete Metric Spaces

Farshid Khojasteh1 Mujahid Abbas2 and Simona Costache3

1 Department of Mathematics Arak Branch Islamic Azad University Arak Iran2Department of Mathematics Lahore University of Management Sciences Lahore 54792 Pakistan3Department of Mathematics and Informatics University Politehnica of Bucharest 060042 Bucharest Romania

Correspondence should be addressed to Simona Costache simona costache2003yahoocom

Received 12 May 2014 Accepted 13 June 2014 Published 26 June 2014

Academic Editor Abdul Latif

Copyright copy 2014 Farshid Khojasteh et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

We introduce two new types of fixed point theorems in the collection ofmultivalued and single-valuedmappings in completemetricspaces

1 Introduction

Let 119879 be a mapping on a complete (or compact) metric space(119883 119889) We do not assume richer structure such as convexmetric spaces and Banach spaces There are thousands oftheorems which assure the existence of a fixed point of 119879 Wecan categorize these theorems into the following four types

(T1) Leader type [1] 119879 has a unique fixed point and 119879119899119909converges to the fixed point for all 119909 isin 119883 Such amapping is called a Picard operator in [2]

(T2) Unnamed type 119879 has a unique fixed point and 119879119899119909does not necessarily converge to the fixed point

(T3) Subrahmanyam type [3] 119879 may have more than onefixed point and 119879119899119909 converges to a fixed point forall 119909 isin 119883 Such a mapping is called a weakly Picardoperator in [3 4]

(T4) Caristi type [5 6] 119879 may have more than one fixedpoint and 119879119899119909 does not necessarily converge to afixed point

We know that most of the theorems such as Banachrsquos[7] Ciricrsquos [8] Kannanrsquos [9] Kirkrsquos [10] Matkowskirsquos [11]Meir and Keelerrsquos [12] and Suzukirsquos [13 14] belong to(1198791) Also very recently Suzuki [15] characterized (1198791)Subrahmanyamrsquos theorem [3] belongs to (1198793) and Caristirsquostheorem [5 6] and its generalizations [15ndash17] belong to (1198794)

On the other hand as far as the authors do know there areno theorems belonging to (1198792) see Kirkrsquos survey [18] Alsorecently many interesting fixed point theorems are provedin the framework of ordered metric spaces see [18ndash35] andothers

In this paper motivated by the above we introducetwo new types of fixed point theorems in the collection ofmultivalued and single-valuedmappings andwill prove themwhich belong to (1198793)

Let (119883 119889) be a metric space and let 119875clbd(119883) denote theclass of all nonempty closed and bounded subsets of 119883 Let119879 119883 rarr 119875clbd(119883) be a multivalued mapping on 119883 A point119909 isin 119883 is called a fixed point of 119879 if 119909 isin 119879119909 Set Fix(119879) = 119909 isin119883 119909 isin 119879119909

A famous theorem on multivalued mappings is due toNadler [36] which extended the Banach contraction princi-ple to multivalued mappings Many authors have studied theexistence anduniqueness of strict fixed points formultivaluedmappings in metric spaces see for example [37ndash44] andreferences therein

Let119867 be the Hausdorff metric on 119875clbd(119883) induced by 119889that is

119867(119860 119861) = maxsup119909isin119861

119889 (119909 119860) sup119909isin119860

119889 (119909 119861)

119860 119861 isin 119875clbd (119883)

(1)

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 325840 5 pageshttpdxdoiorg1011552014325840

2 Abstract and Applied Analysis

Denote 120575(119909 119860) = sup119889(119909 119910) 119910 isin 119860 and 119863(119909 119860) =inf 119889 (119909 119910) 119910 isin 119860 where 119860 isin 119875clbd(119883)

2 Main Results

The following is the first our main results

Theorem 1 Let (119883 119889) be a complete metric space and let 119879be a mapping from 119883 into itself Suppose that 119879 satisfies thefollowing condition

119889 (119879119909 119879119910) le (119889 (119909 119879119910) + 119889 (119910 119879119909)

119889 (119909 119879119909) + 119889 (119910 119879119910) + 1)119889 (119909 119910) (2)

for all 119909 119910 isin 119883 Then

(a) 119879 has at least one fixed point isin 119883(b) 119879119899119909 converges to a fixed point for all 119909 isin 119883(c) if 119910 are two distinct fixed points of 119879 then 119889( 119910) ge12

Proof Let 1199090isin 119883 be arbitrary and choose a sequence 119909

119899

such that 119909119899+1= 119879119909119899 We have

119889 (119909119899+1 119909119899) = 119889 (119879119909

119899 119879119909119899minus1)

le (119889 (119909119899 119909119899) + 119889 (119909

119899minus1 119909119899+1)

119889 (119909119899 119909119899+1) + 119889 (119909

119899minus1 119909119899) + 1

)119889 (119909119899 119909119899minus1)

= (119889 (119909119899minus1 119909119899+1)

119889 (119909119899 119909119899+1) + 119889 (119909

119899minus1 119909119899) + 1

)119889 (119909119899 119909119899minus1)

le(119889 (119909119899minus1 119909119899) + 119889 (119909

119899 119909119899+1)

119889 (119909119899 119909119899+1) + 119889 (119909

119899minus1 119909119899) + 1

)119889 (119909119899 119909119899minus1)

(3)

Given

120573119899=

119889 (119909119899minus1 119909119899) + 119889 (119909

119899 119909119899+1)

119889 (119909119899 119909119899+1) + 119889 (119909

119899minus1 119909119899) + 1

(4)

we have

119889 (119909119899+1 119909119899) le 120573119899119889 (119909119899 119909119899minus1)

le (120573119899120573119899minus1) 119889 (119909119899minus1 119909119899minus2)

le (120573119899120573119899minus1sdot sdot sdot 1205731) 119889 (1199091 1199090)

(5)

Observe that (120573119899) is nonincreasing with positive terms

So 1205731sdot sdot sdot 120573119899le 120573119899

1and 120573119899

1rarr 0 It follows that

lim119899rarrinfin

(12057311205732sdot sdot sdot 120573119899) = 0 (6)

Thus it is verified that

lim119899rarrinfin

119889 (119909119899+1 119909119899) = 0 (7)

Now for all119898 119899 isin N we have

119889 (119909119898 119909119899) le 119889 (119909

119899 119909119899+1) + 119889 (119909

119899+1 119909119899+2)

+ sdot sdot sdot + 119889 (119909119898minus1 119909119898)

le [(120573119899120573119899minus1sdot sdot sdot 1205731) + (120573

119899+1120573119899sdot sdot sdot 1205731)

+ sdot sdot sdot + (120573119898minus1120573119898minus2

sdot sdot sdot 1205731)] 119889 (119909

1 1199090)

=

119898minus1

sum

119896=119899

(120573119896120573119896minus1sdot sdot sdot 1205731) 119889 (1199091 1199090)

(8)

Suppose that 119886119896= (120573119896120573119896minus1sdot sdot sdot 1205731) Since

lim119896rarrinfin

119886119896+1

119886119896

= 0 (9)

suminfin

119896=1119886119896lt infin It means that

119898minus1

sum

119896=119899

(120573119896120573119896minus1sdot sdot sdot 1205731) 997888rarr 0 (10)

as119898 119899 rarr infin In other words 119909119899 is a Cauchy sequence and

so converges to isin 119883We claim that is a fixed pointNote that

119889 (119909119899+1 119879) le (

119889 (119909119899 119879) + 119889 ( 119879119909

119899)

119889 ( 119879) + 119889 (119909119899 119879119909119899) + 1

)119889 (119909119899 )

(11)

On taking limit on both sides of (11) we have 119889( 119879) = 0Thus 119879 =

If there exist two distinct fixed points 119910 isin 119883 then

119889 ( 119910) = 119889 (119879 119879 119910)

le [119889 ( 119879 119910) + 119889 (119879 119910)] 119889 ( 119910)

= 2[119889 ( 119910)]2

(12)

Therefore 119889( 119910) ge 12 and we find the desired results

In the following two examples of such type of mappingswhich satisfy (2) are given

Example 2 Let 119883 = 0 12 1 and let 119889 119883 times 119883 rarr [0infin)

be defined by

119889(01

2) = 2 119889 (1

1

2) =

5

2 119889 (0 1) = 3

119889 (0 0) = 119889 (1

21

2) = 119889 (1 1) = 0

119889 (119886 119887) = 119889 (119887 119886) forall 119886 119887 isin 119883

(13)

Abstract and Applied Analysis 3

(119883 119889) is a complete metric space Let 119879 119883 rarr 119883 be definedby

119879 (0) = 0 119879 (1

2) =

1

2 119879 (1) = 0

119889 (1198790 1198791) = 119889 (0 0) = 0

119889 (1198790 119879 (1

2)) = 119889(0

1

2) = 2

119889 (1198791 119879 (1

2)) = 119889(0

1

2) = 2

(14)

and we have

119889 (1198790 119879 (1

2)) = 119889(0

1

2) = 2

le (119889 (0 119879 (12)) + 119889 (12 119879 (0))

119889 (0 1198790) + 119889 (12 119879 (12)) + 1)

times 119889(01

2) = 8

(15)

and also

119889 (1198791 119879 (1

2)) = 119889(0

1

2) = 2

le (119889 (1 119879 (12)) + 119889 (12 119879 (1))

119889 (1 1198791) + 119889 (12 119879 (12)) + 1)

times 119889(11

2)

= (52 + 2

4) times

5

2=45

16

(16)

Therefore 119879 satisfies all the conditions of Theorem 1 Also 119879has two distinct fixed points 0 12 and 119889(0 12) = 2 ge 12

Example 3 Let 119883 = [0 2 minus radic3] be endowed with Euclideanmetric and let 119879 119883 rarr 119883 be defined by

119879119909 = 0 0 le 119909 lt 2 minus radic3

2 minus radic3 119909 = 2 minus radic3(17)

Thenwe claim that119879 satisfies all the conditions ofTheorem 1If 119909 = 2 minus radic3 and 0 le 119910 lt 2 minus radic3 we have

1003816100381610038161003816119879119909 minus 1198791199101003816100381610038161003816 (|119909 minus 119879119909| +

1003816100381610038161003816119910 minus 1198791199101003816100381610038161003816 + 1)

= (2 minus radic3) (10038161003816100381610038161199101003816100381610038161003816 + 1) = (2 minus

radic3) (119910 + 1)

le (2 minus radic3 minus 119910)2

minus (2 minus radic3) (2 minus radic3 minus 119910)

= (1003816100381610038161003816119909 minus 119879119910

1003816100381610038161003816 +1003816100381610038161003816119910 minus 119879119909

1003816100381610038161003816)1003816100381610038161003816119909 minus 119910

1003816100381610038161003816

(18)

Thus

1003816100381610038161003816119879119909 minus 1198791199101003816100381610038161003816 le (

1003816100381610038161003816119909 minus 1198791199101003816100381610038161003816 +1003816100381610038161003816119910 minus 119879119909

1003816100381610038161003816

|119909 minus 119879119909| +1003816100381610038161003816119910 minus 119879119910

1003816100381610038161003816 + 1)1003816100381610038161003816119909 minus 119910

1003816100381610038161003816 (19)

Similar argument holds for the other conditions

Remark 4 Note that in (2) the ratio

119889 (119909 119879119910) + 119889 (119910 119879119909)

119889 (119909 119879119909) + 119889 (119910 119879119910) + 1(20)

might be greater or less than 1 and has not introduced anupper bound Note that if for every 119909 119910 isin 119883 119889(119909 119910) lt 12then we have

119889 (119909 119879119910) + 119889 (119910 119879119909)

le 2119889 (119909 119910) + 119889 (119909 119879119909) + 119889 (119910 119879119910)

lt 119889 (119909 119879119909) + 119889 (119910 119879119910) + 1

(21)

It means that

119889 (119909 119879119910) + 119889 (119910 119879119909)

119889 (119909 119879119909) + 119889 (119910 119879119910) + 1lt 1 (22)

and thus Theorem 1 is a special case of Banach contractionprinciple Therefore when (119883 119889) is a complete metric spacesuch that for all 119909 119910 isin 119883 119889(119909 119910) ge 12 Theorem 1 isvaluable because (20) might be greater than 1 Example 2shows this note precisely

The following is the second in our main results

Theorem 5 Let (119883 119889) be a complete metric space and let 119879 bea multivalued mapping from119883 into 119875

119888119897119887119889(119883) Let 119879 satisfy the

following

119867(119879119909 119879119910) le (119863 (119909 119879119910) + 119863 (119910 119879119909)

120575 (119909 119879119909) + 120575 (119910 119879119910) + 1)119889 (119909 119910) (23)

for all 119909 119910 isin 119883 Then 119879 has a fixed point isin 119883

Proof Let 1199090isin 119883 and 119909

1isin 1198791199090 For each 0 lt ℎ

1lt 1 one can

choose 1199092isin 1198791199091such that

119889 (1199091 1199092) lt 119867 (119879119909

0 1198791199091) + (1 minus

1

ℎ1

)119867(1198791199090 1198791199091)

=1

ℎ1

119867(1198791199090 1198791199091)

(24)

For each 0 lt ℎ119899lt 1 we can choose 119909

119899+1isin 119879119909119899such that

119889 (119909119899 119909119899+1) lt 119867 (119879119909

119899minus1 119879119909119899) + (1 minus

1

ℎ119899

)119867(1198791199090 1198791199091)

=1

ℎ119899

119867(1198791199090 1198791199091)

(25)

Specifically if

ℎ119899= radic

119889 (119909119899minus1+ 119909119899+1)

119889 (119909119899minus1+ 119909119899) + 119889 (119909

119899+ 119909119899+1) + 1

= radic120573119899

(26)

4 Abstract and Applied Analysis

then

119889 (119909119899 119909119899+1) le radic120573

119899119889 (119909119899minus1 119909119899) le 120573119899119889 (119909119899minus1 119909119899) (27)

Therefore119889 (119909119899+1 119909119899) le 120573119899119889 (119909119899 119909119899minus1)

le (120573119899120573119899minus1) 119889 (119909119899minus1 119909119899minus2)

le (120573119899120573119899minus1sdot sdot sdot 1205731) 119889 (1199091 1199090)

(28)

It can easily be seen that

lim119899rarrinfin

(12057311205732sdot sdot sdot 120573119899) = 0 (29)

Thus it is easily verified that

lim119899rarrinfin

119889 (119909119899+1 119909119899) = 0 (30)

Now for all119898 119899 isin N we have119889 (119909119898 119909119899) le 119889 (119909

119899 119909119899+1) + 119889 (119909

119899 119909119899+1) + sdot sdot sdot + 119889 (119909

119898minus1 119909119898)

le [(120573119899120573119899minus1sdot sdot sdot 1205731) + (120573

119899+1120573119899sdot sdot sdot 1205731)

+ sdot sdot sdot + (120573119898minus1120573119898minus2

sdot sdot sdot 1205731)] 119889 (119909

1 1199090)

=

119898minus1

sum

119896=119899

(120573119896120573119896minus1sdot sdot sdot 1205731) 119889 (1199091 1199090)

(31)

Suppose that 119886119896= (120573119896120573119896minus1sdot sdot sdot 1205731) Since

lim119896rarrinfin

119886119896+1

119886119896

= 0 (32)

suminfin

119896=1119886119896lt infin It means that

119898minus1

sum

119896=119899

(120573119896120573119896minus1sdot sdot sdot 1205731) 997888rarr 0 (33)

as 119898 119899 rarr infin In other words 119909119899 is a Cauchy sequence

and so converges to isin 119883 We claim that is a fixed pointConsider

119863 ( 119879) le 119889 ( 119909119899+1) + 119863 (119909

119899+1 119879)

le 119867 (119879119909119899 119879) + 119889 ( 119909

119899+1)

le (119863 ( 119879119909

119899) + 119863 (119909

119899 119879)

120575 ( 119879) + 120575 (119909119899 119879119909119899) + 1

)

times 119889 (119909119899 ) + 119889 ( 119909

119899+1)

le [119863 ( 119909119899+1) + 119863 (119909

119899 119879)]

times 119889 (119909119899 ) + 119889 ( 119909

119899+1)

(34)

On taking limit on both sides of (31) we have 119863( 119879) = 0It means that isin 119879

Remark 6 Note that Theorem 5 is a generalization ofTheorem 1 because by taking 119865119909 = 119879119909 and applyingTheorem 5 for 119865 we obtainTheorem 1

Conflict of Interests

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

References

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[2] I A Rus ldquoPicard operators and applicationsrdquo Scientiae Mathe-maticae vol 58 no 1 pp 191ndash219 2003

[3] P V Subrahmanyam ldquoRemarks on some fixed point theoremsrelated to Banachrsquos contraction principlerdquo Journal of Mathemat-ical and Physical Sciences vol 8 pp 445ndash457 1974

[4] I A Rus A S Muresan and V Muresan ldquoWeakly Picardoperators on a set with two metricsrdquo Fixed Point Theory vol6 no 2 pp 323ndash331 2005

[5] J Caristi ldquoFixed point theorems for mappings satisfyinginwardness conditionsrdquo Transactions of the American Mathe-matical Society vol 215 pp 241ndash251 1976

[6] J Caristi and W A Kirk ldquoGeometric fixed point theory andinwardness conditionsrdquo in The Geometry of Metric and LinearSpaces vol 490 of Lecture Notes in Mathematics pp 74ndash83Springer Berlin Germany 1975

[7] S Banach ldquoSur les operations dans les ensembles abstraits etleur application aux equations integralesrdquo Fundamenta Mathe-maticae vol 3 pp 133ndash181 1922

[8] L B Ciric ldquoA generalization of Banachrsquos contraction principlerdquoProceedings of the American Mathematical Society vol 45 pp267ndash273 1974

[9] R Kannan ldquoSome results on fixed pointsmdashIIrdquo The AmericanMathematical Monthly vol 76 no 4 pp 405ndash408 1969

[10] W A Kirk ldquoFixed points of asymptotic contractionsrdquo Journalof Mathematical Analysis and Applications vol 277 no 2 pp645ndash650 2003

[11] J Matkowski ldquoIntegrable solutions of functional equationsrdquoDissertationes Mathematicae vol 127 pp 1ndash68 1975

[12] A Meir and E Keeler ldquoA theorem on contraction mappingsrdquoJournal of Mathematical Analysis and Applications vol 28 pp326ndash329 1969

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

[14] T Suzuki ldquoSeveral fixed point theorems concerning 120591-distancerdquoFixed Point Theory and Applications vol 2004 no 3 pp 195ndash209 2004

[15] T Suzuki ldquoA sufficient and necessary condition for the conver-gence of the sequence of successive approximations to a uniquefixed pointrdquo Proceedings of the American Mathematical Societyvol 136 no 11 pp 4089ndash4093 2008

[16] J S Bae ldquoFixed point theorems for weakly contractivemultival-ued mapsrdquo Journal of Mathematical Analysis and Applicationsvol 284 no 2 pp 690ndash697 2003

[17] J S Bae E W Cho and S H Yeom ldquoA generalization ofthe Caristi-Kirk fixed point theorem and its applications tomapping theoremsrdquo Journal of the KoreanMathematical Societyvol 31 no 1 pp 29ndash48 1994

[18] W A Kirk ldquoContraction mappings and extensionsrdquo in Hand-book of Metric Fixed PointTheory W A Kirk and B Sims Edspp 1ndash34 Kluwer Academic Dordrecht The Netherlands 2001

Abstract and Applied Analysis 5

[19] H AydiW ShatanawiM Postolache ZMustafa andN TahatldquoTheorems for Boyd-Wong-type contractions in orderedmetricspacesrdquo Abstract and Applied Analysis vol 2012 Article ID359054 14 pages 2012

[20] S Chandok andM Postolache ldquoFixed point theorem forweaklyChatterjea-type cyclic contractionsrdquo Fixed Point Theory andApplications vol 2013 no 28 2013

[21] S Chandok Z Mustafa andM Postolache ldquoCoupled commonfixed point results for mixed g-monotone mapps in partiallyordered G-metric spacesrdquo University Politehnica of BucharestScientific Bulletin A vol 75 no 4 pp 13ndash26 2013

[22] Y Chen ldquoStability of positive fixed points of nonlinear opera-torsrdquo Positivity vol 6 no 1 pp 47ndash57 2002

[23] B S Choudhury N Metiya and M Postolache ldquoA generalizedweak contraction principle with applications to coupled coin-cidence point problemsrdquo Fixed Point Theory and Applicationsvol 2013 article 152 2013

[24] R H Haghi M Postolache and S Rezapour ldquoOn T-stabilityof the Picard iteration for generalized 120593-contractionmappingsrdquoAbstract and Applied Analysis vol 2012 Article ID 658971 7pages 2012

[25] D Ilic and V Rakocevic ldquoCommon fixed points for mapson cone metric spacerdquo Journal of Mathematical Analysis andApplications vol 341 no 2 pp 876ndash882 2008

[26] J Jachymski ldquoThe contraction principle for mappings ona metric space with a graphrdquo Proceedings of the AmericanMathematical Society vol 136 no 4 pp 1359ndash1373 2008

[27] V Lakshmikantham and L Ciric ldquoCoupled fixed point the-orems for nonlinear contractions in partially ordered metricspacesrdquoNonlinearAnalysisTheoryMethodsampApplications vol70 no 12 pp 4341ndash4349 2009

[28] J J Nieto R L Pouso and R Rodriguez-Lopez ldquoFixedpoint theorems in ordered abstract spacesrdquo Proceedings of theAmerican Mathematical Society vol 135 no 8 pp 2505ndash25172007

[29] J J Nieto and R Rodriguez-Lopez ldquoContractive mappingtheorems in partially ordered sets and applications to ordinarydifferential equationsrdquo Order vol 22 no 3 pp 223ndash239 2005

[30] J J Nieto and R Rodrıguez-Lopez ldquoExistence and uniquenessof fixed point in partially ordered sets and applications toordinary differential equationsrdquo Acta Mathematica Sinica vol23 no 12 pp 2205ndash2212 2007

[31] D OrsquoRegan and A Petrusel ldquoFixed point theorems for gen-eralized contractions in ordered metric spacesrdquo Journal ofMathematical Analysis andApplications vol 341 no 2 pp 1241ndash1252 2008

[32] H K Pathak and N Shahzad ldquoFixed points for generalizedcontractions and applications to control theoryrdquo NonlinearAnalysisTheoryMethodsampApplications vol 68 no 8 pp 2181ndash2193 2008

[33] W Shatanawi and M Postolache ldquoCommon fixed point theo-rems for dominating andweak annihilatormappings in orderedmetric spacesrdquo Fixed Point Theory and Applications vol 2013article 271 2013

[34] W Shatanawi and M Postolache ldquoCommon fixed point resultsfor mappings under nonlinear contraction of cyclic form inorderedmetric spacesrdquo Fixed PointTheory andApplications vol2013 no 60 2013

[35] Y Wu ldquoNew fixed point theorems and applications of mixedmonotone operatorrdquo Journal of Mathematical Analysis andApplications vol 341 no 2 pp 883ndash893 2008

[36] J Nadler ldquoMulti-valued contraction mappingsrdquo Pacific Journalof Mathematics vol 30 pp 475ndash488 1969

[37] A Amini-Harandi ldquoEndpoints of set-valued contractions inmetric spacesrdquo Nonlinear Analysis Theory Methods amp Applica-tions vol 72 no 1 pp 132ndash134 2010

[38] A Amini-Harandi ldquoFixed point theory for set-valued quasi-contraction maps in metric spacesrdquo Applied Mathematics Let-ters vol 24 no 11 pp 1791ndash1794 2011

[39] L B Ciric and J S Ume ldquoMulti-valued non-self-mappings onconvex metric spacesrdquo Nonlinear Analysis Theory Methods ampApplications vol 60 no 6 pp 1053ndash1063 2005

[40] M Fakhar ldquoEndpoints of set-valued asymptotic contractions inmetric spacesrdquo Applied Mathematics Letters vol 24 no 4 pp428ndash431 2011

[41] N Hussain A Amini-Harandi and Y J Cho ldquoApproximateendpoints for set-valued contractions in metric spacesrdquo FixedPoint Theory and Applications vol 2010 Article ID 614867 13pages 2010

[42] Z Kadelburg and S Radenovic ldquoSome results on set-valuedcontractions in abstract metric spacesrdquo Computers amp Mathe-matics with Applications vol 62 no 1 pp 342ndash350 2011

[43] S Moradi and F Khojasteh ldquoEndpoints of multi-valued gener-alized weak contraction mappingsrdquo Nonlinear Analysis TheoryMethods amp Applications vol 74 no 6 pp 2170ndash2174 2011

[44] F Khojasteh and Rakocevic V ldquoSome new common fixedpoint results for generalized contractive multi-valued non-self-mappingsrdquo Applied Mathematics Letters vol 25 no 3 pp 287ndash293 2012

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Abstract and Applied Analysis

Denote 120575(119909 119860) = sup119889(119909 119910) 119910 isin 119860 and 119863(119909 119860) =inf 119889 (119909 119910) 119910 isin 119860 where 119860 isin 119875clbd(119883)

2 Main Results

The following is the first our main results

Theorem 1 Let (119883 119889) be a complete metric space and let 119879be a mapping from 119883 into itself Suppose that 119879 satisfies thefollowing condition

119889 (119879119909 119879119910) le (119889 (119909 119879119910) + 119889 (119910 119879119909)

119889 (119909 119879119909) + 119889 (119910 119879119910) + 1)119889 (119909 119910) (2)

for all 119909 119910 isin 119883 Then

(a) 119879 has at least one fixed point isin 119883(b) 119879119899119909 converges to a fixed point for all 119909 isin 119883(c) if 119910 are two distinct fixed points of 119879 then 119889( 119910) ge12

Proof Let 1199090isin 119883 be arbitrary and choose a sequence 119909

119899

such that 119909119899+1= 119879119909119899 We have

119889 (119909119899+1 119909119899) = 119889 (119879119909

119899 119879119909119899minus1)

le (119889 (119909119899 119909119899) + 119889 (119909

119899minus1 119909119899+1)

119889 (119909119899 119909119899+1) + 119889 (119909

119899minus1 119909119899) + 1

)119889 (119909119899 119909119899minus1)

= (119889 (119909119899minus1 119909119899+1)

119889 (119909119899 119909119899+1) + 119889 (119909

119899minus1 119909119899) + 1

)119889 (119909119899 119909119899minus1)

le(119889 (119909119899minus1 119909119899) + 119889 (119909

119899 119909119899+1)

119889 (119909119899 119909119899+1) + 119889 (119909

119899minus1 119909119899) + 1

)119889 (119909119899 119909119899minus1)

(3)

Given

120573119899=

119889 (119909119899minus1 119909119899) + 119889 (119909

119899 119909119899+1)

119889 (119909119899 119909119899+1) + 119889 (119909

119899minus1 119909119899) + 1

(4)

we have

119889 (119909119899+1 119909119899) le 120573119899119889 (119909119899 119909119899minus1)

le (120573119899120573119899minus1) 119889 (119909119899minus1 119909119899minus2)

le (120573119899120573119899minus1sdot sdot sdot 1205731) 119889 (1199091 1199090)

(5)

Observe that (120573119899) is nonincreasing with positive terms

So 1205731sdot sdot sdot 120573119899le 120573119899

1and 120573119899

1rarr 0 It follows that

lim119899rarrinfin

(12057311205732sdot sdot sdot 120573119899) = 0 (6)

Thus it is verified that

lim119899rarrinfin

119889 (119909119899+1 119909119899) = 0 (7)

Now for all119898 119899 isin N we have

119889 (119909119898 119909119899) le 119889 (119909

119899 119909119899+1) + 119889 (119909

119899+1 119909119899+2)

+ sdot sdot sdot + 119889 (119909119898minus1 119909119898)

le [(120573119899120573119899minus1sdot sdot sdot 1205731) + (120573

119899+1120573119899sdot sdot sdot 1205731)

+ sdot sdot sdot + (120573119898minus1120573119898minus2

sdot sdot sdot 1205731)] 119889 (119909

1 1199090)

=

119898minus1

sum

119896=119899

(120573119896120573119896minus1sdot sdot sdot 1205731) 119889 (1199091 1199090)

(8)

Suppose that 119886119896= (120573119896120573119896minus1sdot sdot sdot 1205731) Since

lim119896rarrinfin

119886119896+1

119886119896

= 0 (9)

suminfin

119896=1119886119896lt infin It means that

119898minus1

sum

119896=119899

(120573119896120573119896minus1sdot sdot sdot 1205731) 997888rarr 0 (10)

as119898 119899 rarr infin In other words 119909119899 is a Cauchy sequence and

so converges to isin 119883We claim that is a fixed pointNote that

119889 (119909119899+1 119879) le (

119889 (119909119899 119879) + 119889 ( 119879119909

119899)

119889 ( 119879) + 119889 (119909119899 119879119909119899) + 1

)119889 (119909119899 )

(11)

On taking limit on both sides of (11) we have 119889( 119879) = 0Thus 119879 =

If there exist two distinct fixed points 119910 isin 119883 then

119889 ( 119910) = 119889 (119879 119879 119910)

le [119889 ( 119879 119910) + 119889 (119879 119910)] 119889 ( 119910)

= 2[119889 ( 119910)]2

(12)

Therefore 119889( 119910) ge 12 and we find the desired results

In the following two examples of such type of mappingswhich satisfy (2) are given

Example 2 Let 119883 = 0 12 1 and let 119889 119883 times 119883 rarr [0infin)

be defined by

119889(01

2) = 2 119889 (1

1

2) =

5

2 119889 (0 1) = 3

119889 (0 0) = 119889 (1

21

2) = 119889 (1 1) = 0

119889 (119886 119887) = 119889 (119887 119886) forall 119886 119887 isin 119883

(13)

Abstract and Applied Analysis 3

(119883 119889) is a complete metric space Let 119879 119883 rarr 119883 be definedby

119879 (0) = 0 119879 (1

2) =

1

2 119879 (1) = 0

119889 (1198790 1198791) = 119889 (0 0) = 0

119889 (1198790 119879 (1

2)) = 119889(0

1

2) = 2

119889 (1198791 119879 (1

2)) = 119889(0

1

2) = 2

(14)

and we have

119889 (1198790 119879 (1

2)) = 119889(0

1

2) = 2

le (119889 (0 119879 (12)) + 119889 (12 119879 (0))

119889 (0 1198790) + 119889 (12 119879 (12)) + 1)

times 119889(01

2) = 8

(15)

and also

119889 (1198791 119879 (1

2)) = 119889(0

1

2) = 2

le (119889 (1 119879 (12)) + 119889 (12 119879 (1))

119889 (1 1198791) + 119889 (12 119879 (12)) + 1)

times 119889(11

2)

= (52 + 2

4) times

5

2=45

16

(16)

Therefore 119879 satisfies all the conditions of Theorem 1 Also 119879has two distinct fixed points 0 12 and 119889(0 12) = 2 ge 12

Example 3 Let 119883 = [0 2 minus radic3] be endowed with Euclideanmetric and let 119879 119883 rarr 119883 be defined by

119879119909 = 0 0 le 119909 lt 2 minus radic3

2 minus radic3 119909 = 2 minus radic3(17)

Thenwe claim that119879 satisfies all the conditions ofTheorem 1If 119909 = 2 minus radic3 and 0 le 119910 lt 2 minus radic3 we have

1003816100381610038161003816119879119909 minus 1198791199101003816100381610038161003816 (|119909 minus 119879119909| +

1003816100381610038161003816119910 minus 1198791199101003816100381610038161003816 + 1)

= (2 minus radic3) (10038161003816100381610038161199101003816100381610038161003816 + 1) = (2 minus

radic3) (119910 + 1)

le (2 minus radic3 minus 119910)2

minus (2 minus radic3) (2 minus radic3 minus 119910)

= (1003816100381610038161003816119909 minus 119879119910

1003816100381610038161003816 +1003816100381610038161003816119910 minus 119879119909

1003816100381610038161003816)1003816100381610038161003816119909 minus 119910

1003816100381610038161003816

(18)

Thus

1003816100381610038161003816119879119909 minus 1198791199101003816100381610038161003816 le (

1003816100381610038161003816119909 minus 1198791199101003816100381610038161003816 +1003816100381610038161003816119910 minus 119879119909

1003816100381610038161003816

|119909 minus 119879119909| +1003816100381610038161003816119910 minus 119879119910

1003816100381610038161003816 + 1)1003816100381610038161003816119909 minus 119910

1003816100381610038161003816 (19)

Similar argument holds for the other conditions

Remark 4 Note that in (2) the ratio

119889 (119909 119879119910) + 119889 (119910 119879119909)

119889 (119909 119879119909) + 119889 (119910 119879119910) + 1(20)

might be greater or less than 1 and has not introduced anupper bound Note that if for every 119909 119910 isin 119883 119889(119909 119910) lt 12then we have

119889 (119909 119879119910) + 119889 (119910 119879119909)

le 2119889 (119909 119910) + 119889 (119909 119879119909) + 119889 (119910 119879119910)

lt 119889 (119909 119879119909) + 119889 (119910 119879119910) + 1

(21)

It means that

119889 (119909 119879119910) + 119889 (119910 119879119909)

119889 (119909 119879119909) + 119889 (119910 119879119910) + 1lt 1 (22)

and thus Theorem 1 is a special case of Banach contractionprinciple Therefore when (119883 119889) is a complete metric spacesuch that for all 119909 119910 isin 119883 119889(119909 119910) ge 12 Theorem 1 isvaluable because (20) might be greater than 1 Example 2shows this note precisely

The following is the second in our main results

Theorem 5 Let (119883 119889) be a complete metric space and let 119879 bea multivalued mapping from119883 into 119875

119888119897119887119889(119883) Let 119879 satisfy the

following

119867(119879119909 119879119910) le (119863 (119909 119879119910) + 119863 (119910 119879119909)

120575 (119909 119879119909) + 120575 (119910 119879119910) + 1)119889 (119909 119910) (23)

for all 119909 119910 isin 119883 Then 119879 has a fixed point isin 119883

Proof Let 1199090isin 119883 and 119909

1isin 1198791199090 For each 0 lt ℎ

1lt 1 one can

choose 1199092isin 1198791199091such that

119889 (1199091 1199092) lt 119867 (119879119909

0 1198791199091) + (1 minus

1

ℎ1

)119867(1198791199090 1198791199091)

=1

ℎ1

119867(1198791199090 1198791199091)

(24)

For each 0 lt ℎ119899lt 1 we can choose 119909

119899+1isin 119879119909119899such that

119889 (119909119899 119909119899+1) lt 119867 (119879119909

119899minus1 119879119909119899) + (1 minus

1

ℎ119899

)119867(1198791199090 1198791199091)

=1

ℎ119899

119867(1198791199090 1198791199091)

(25)

Specifically if

ℎ119899= radic

119889 (119909119899minus1+ 119909119899+1)

119889 (119909119899minus1+ 119909119899) + 119889 (119909

119899+ 119909119899+1) + 1

= radic120573119899

(26)

4 Abstract and Applied Analysis

then

119889 (119909119899 119909119899+1) le radic120573

119899119889 (119909119899minus1 119909119899) le 120573119899119889 (119909119899minus1 119909119899) (27)

Therefore119889 (119909119899+1 119909119899) le 120573119899119889 (119909119899 119909119899minus1)

le (120573119899120573119899minus1) 119889 (119909119899minus1 119909119899minus2)

le (120573119899120573119899minus1sdot sdot sdot 1205731) 119889 (1199091 1199090)

(28)

It can easily be seen that

lim119899rarrinfin

(12057311205732sdot sdot sdot 120573119899) = 0 (29)

Thus it is easily verified that

lim119899rarrinfin

119889 (119909119899+1 119909119899) = 0 (30)

Now for all119898 119899 isin N we have119889 (119909119898 119909119899) le 119889 (119909

119899 119909119899+1) + 119889 (119909

119899 119909119899+1) + sdot sdot sdot + 119889 (119909

119898minus1 119909119898)

le [(120573119899120573119899minus1sdot sdot sdot 1205731) + (120573

119899+1120573119899sdot sdot sdot 1205731)

+ sdot sdot sdot + (120573119898minus1120573119898minus2

sdot sdot sdot 1205731)] 119889 (119909

1 1199090)

=

119898minus1

sum

119896=119899

(120573119896120573119896minus1sdot sdot sdot 1205731) 119889 (1199091 1199090)

(31)

Suppose that 119886119896= (120573119896120573119896minus1sdot sdot sdot 1205731) Since

lim119896rarrinfin

119886119896+1

119886119896

= 0 (32)

suminfin

119896=1119886119896lt infin It means that

119898minus1

sum

119896=119899

(120573119896120573119896minus1sdot sdot sdot 1205731) 997888rarr 0 (33)

as 119898 119899 rarr infin In other words 119909119899 is a Cauchy sequence

and so converges to isin 119883 We claim that is a fixed pointConsider

119863 ( 119879) le 119889 ( 119909119899+1) + 119863 (119909

119899+1 119879)

le 119867 (119879119909119899 119879) + 119889 ( 119909

119899+1)

le (119863 ( 119879119909

119899) + 119863 (119909

119899 119879)

120575 ( 119879) + 120575 (119909119899 119879119909119899) + 1

)

times 119889 (119909119899 ) + 119889 ( 119909

119899+1)

le [119863 ( 119909119899+1) + 119863 (119909

119899 119879)]

times 119889 (119909119899 ) + 119889 ( 119909

119899+1)

(34)

On taking limit on both sides of (31) we have 119863( 119879) = 0It means that isin 119879

Remark 6 Note that Theorem 5 is a generalization ofTheorem 1 because by taking 119865119909 = 119879119909 and applyingTheorem 5 for 119865 we obtainTheorem 1

Conflict of Interests

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

References

[1] S Leader ldquoEquivalent Cauchy sequences and contractive fixedpoints in metric spacesrdquo Studia Mathematica vol 76 no 1 pp63ndash67 1983

[2] I A Rus ldquoPicard operators and applicationsrdquo Scientiae Mathe-maticae vol 58 no 1 pp 191ndash219 2003

[3] P V Subrahmanyam ldquoRemarks on some fixed point theoremsrelated to Banachrsquos contraction principlerdquo Journal of Mathemat-ical and Physical Sciences vol 8 pp 445ndash457 1974

[4] I A Rus A S Muresan and V Muresan ldquoWeakly Picardoperators on a set with two metricsrdquo Fixed Point Theory vol6 no 2 pp 323ndash331 2005

[5] J Caristi ldquoFixed point theorems for mappings satisfyinginwardness conditionsrdquo Transactions of the American Mathe-matical Society vol 215 pp 241ndash251 1976

[6] J Caristi and W A Kirk ldquoGeometric fixed point theory andinwardness conditionsrdquo in The Geometry of Metric and LinearSpaces vol 490 of Lecture Notes in Mathematics pp 74ndash83Springer Berlin Germany 1975

[7] S Banach ldquoSur les operations dans les ensembles abstraits etleur application aux equations integralesrdquo Fundamenta Mathe-maticae vol 3 pp 133ndash181 1922

[8] L B Ciric ldquoA generalization of Banachrsquos contraction principlerdquoProceedings of the American Mathematical Society vol 45 pp267ndash273 1974

[9] R Kannan ldquoSome results on fixed pointsmdashIIrdquo The AmericanMathematical Monthly vol 76 no 4 pp 405ndash408 1969

[10] W A Kirk ldquoFixed points of asymptotic contractionsrdquo Journalof Mathematical Analysis and Applications vol 277 no 2 pp645ndash650 2003

[11] J Matkowski ldquoIntegrable solutions of functional equationsrdquoDissertationes Mathematicae vol 127 pp 1ndash68 1975

[12] A Meir and E Keeler ldquoA theorem on contraction mappingsrdquoJournal of Mathematical Analysis and Applications vol 28 pp326ndash329 1969

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

[14] T Suzuki ldquoSeveral fixed point theorems concerning 120591-distancerdquoFixed Point Theory and Applications vol 2004 no 3 pp 195ndash209 2004

[15] T Suzuki ldquoA sufficient and necessary condition for the conver-gence of the sequence of successive approximations to a uniquefixed pointrdquo Proceedings of the American Mathematical Societyvol 136 no 11 pp 4089ndash4093 2008

[16] J S Bae ldquoFixed point theorems for weakly contractivemultival-ued mapsrdquo Journal of Mathematical Analysis and Applicationsvol 284 no 2 pp 690ndash697 2003

[17] J S Bae E W Cho and S H Yeom ldquoA generalization ofthe Caristi-Kirk fixed point theorem and its applications tomapping theoremsrdquo Journal of the KoreanMathematical Societyvol 31 no 1 pp 29ndash48 1994

[18] W A Kirk ldquoContraction mappings and extensionsrdquo in Hand-book of Metric Fixed PointTheory W A Kirk and B Sims Edspp 1ndash34 Kluwer Academic Dordrecht The Netherlands 2001

Abstract and Applied Analysis 5

[19] H AydiW ShatanawiM Postolache ZMustafa andN TahatldquoTheorems for Boyd-Wong-type contractions in orderedmetricspacesrdquo Abstract and Applied Analysis vol 2012 Article ID359054 14 pages 2012

[20] S Chandok andM Postolache ldquoFixed point theorem forweaklyChatterjea-type cyclic contractionsrdquo Fixed Point Theory andApplications vol 2013 no 28 2013

[21] S Chandok Z Mustafa andM Postolache ldquoCoupled commonfixed point results for mixed g-monotone mapps in partiallyordered G-metric spacesrdquo University Politehnica of BucharestScientific Bulletin A vol 75 no 4 pp 13ndash26 2013

[22] Y Chen ldquoStability of positive fixed points of nonlinear opera-torsrdquo Positivity vol 6 no 1 pp 47ndash57 2002

[23] B S Choudhury N Metiya and M Postolache ldquoA generalizedweak contraction principle with applications to coupled coin-cidence point problemsrdquo Fixed Point Theory and Applicationsvol 2013 article 152 2013

[24] R H Haghi M Postolache and S Rezapour ldquoOn T-stabilityof the Picard iteration for generalized 120593-contractionmappingsrdquoAbstract and Applied Analysis vol 2012 Article ID 658971 7pages 2012

[25] D Ilic and V Rakocevic ldquoCommon fixed points for mapson cone metric spacerdquo Journal of Mathematical Analysis andApplications vol 341 no 2 pp 876ndash882 2008

[26] J Jachymski ldquoThe contraction principle for mappings ona metric space with a graphrdquo Proceedings of the AmericanMathematical Society vol 136 no 4 pp 1359ndash1373 2008

[27] V Lakshmikantham and L Ciric ldquoCoupled fixed point the-orems for nonlinear contractions in partially ordered metricspacesrdquoNonlinearAnalysisTheoryMethodsampApplications vol70 no 12 pp 4341ndash4349 2009

[28] J J Nieto R L Pouso and R Rodriguez-Lopez ldquoFixedpoint theorems in ordered abstract spacesrdquo Proceedings of theAmerican Mathematical Society vol 135 no 8 pp 2505ndash25172007

[29] J J Nieto and R Rodriguez-Lopez ldquoContractive mappingtheorems in partially ordered sets and applications to ordinarydifferential equationsrdquo Order vol 22 no 3 pp 223ndash239 2005

[30] J J Nieto and R Rodrıguez-Lopez ldquoExistence and uniquenessof fixed point in partially ordered sets and applications toordinary differential equationsrdquo Acta Mathematica Sinica vol23 no 12 pp 2205ndash2212 2007

[31] D OrsquoRegan and A Petrusel ldquoFixed point theorems for gen-eralized contractions in ordered metric spacesrdquo Journal ofMathematical Analysis andApplications vol 341 no 2 pp 1241ndash1252 2008

[32] H K Pathak and N Shahzad ldquoFixed points for generalizedcontractions and applications to control theoryrdquo NonlinearAnalysisTheoryMethodsampApplications vol 68 no 8 pp 2181ndash2193 2008

[33] W Shatanawi and M Postolache ldquoCommon fixed point theo-rems for dominating andweak annihilatormappings in orderedmetric spacesrdquo Fixed Point Theory and Applications vol 2013article 271 2013

[34] W Shatanawi and M Postolache ldquoCommon fixed point resultsfor mappings under nonlinear contraction of cyclic form inorderedmetric spacesrdquo Fixed PointTheory andApplications vol2013 no 60 2013

[35] Y Wu ldquoNew fixed point theorems and applications of mixedmonotone operatorrdquo Journal of Mathematical Analysis andApplications vol 341 no 2 pp 883ndash893 2008

[36] J Nadler ldquoMulti-valued contraction mappingsrdquo Pacific Journalof Mathematics vol 30 pp 475ndash488 1969

[37] A Amini-Harandi ldquoEndpoints of set-valued contractions inmetric spacesrdquo Nonlinear Analysis Theory Methods amp Applica-tions vol 72 no 1 pp 132ndash134 2010

[38] A Amini-Harandi ldquoFixed point theory for set-valued quasi-contraction maps in metric spacesrdquo Applied Mathematics Let-ters vol 24 no 11 pp 1791ndash1794 2011

[39] L B Ciric and J S Ume ldquoMulti-valued non-self-mappings onconvex metric spacesrdquo Nonlinear Analysis Theory Methods ampApplications vol 60 no 6 pp 1053ndash1063 2005

[40] M Fakhar ldquoEndpoints of set-valued asymptotic contractions inmetric spacesrdquo Applied Mathematics Letters vol 24 no 4 pp428ndash431 2011

[41] N Hussain A Amini-Harandi and Y J Cho ldquoApproximateendpoints for set-valued contractions in metric spacesrdquo FixedPoint Theory and Applications vol 2010 Article ID 614867 13pages 2010

[42] Z Kadelburg and S Radenovic ldquoSome results on set-valuedcontractions in abstract metric spacesrdquo Computers amp Mathe-matics with Applications vol 62 no 1 pp 342ndash350 2011

[43] S Moradi and F Khojasteh ldquoEndpoints of multi-valued gener-alized weak contraction mappingsrdquo Nonlinear Analysis TheoryMethods amp Applications vol 74 no 6 pp 2170ndash2174 2011

[44] F Khojasteh and Rakocevic V ldquoSome new common fixedpoint results for generalized contractive multi-valued non-self-mappingsrdquo Applied Mathematics Letters vol 25 no 3 pp 287ndash293 2012

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 3

(119883 119889) is a complete metric space Let 119879 119883 rarr 119883 be definedby

119879 (0) = 0 119879 (1

2) =

1

2 119879 (1) = 0

119889 (1198790 1198791) = 119889 (0 0) = 0

119889 (1198790 119879 (1

2)) = 119889(0

1

2) = 2

119889 (1198791 119879 (1

2)) = 119889(0

1

2) = 2

(14)

and we have

119889 (1198790 119879 (1

2)) = 119889(0

1

2) = 2

le (119889 (0 119879 (12)) + 119889 (12 119879 (0))

119889 (0 1198790) + 119889 (12 119879 (12)) + 1)

times 119889(01

2) = 8

(15)

and also

119889 (1198791 119879 (1

2)) = 119889(0

1

2) = 2

le (119889 (1 119879 (12)) + 119889 (12 119879 (1))

119889 (1 1198791) + 119889 (12 119879 (12)) + 1)

times 119889(11

2)

= (52 + 2

4) times

5

2=45

16

(16)

Therefore 119879 satisfies all the conditions of Theorem 1 Also 119879has two distinct fixed points 0 12 and 119889(0 12) = 2 ge 12

Example 3 Let 119883 = [0 2 minus radic3] be endowed with Euclideanmetric and let 119879 119883 rarr 119883 be defined by

119879119909 = 0 0 le 119909 lt 2 minus radic3

2 minus radic3 119909 = 2 minus radic3(17)

Thenwe claim that119879 satisfies all the conditions ofTheorem 1If 119909 = 2 minus radic3 and 0 le 119910 lt 2 minus radic3 we have

1003816100381610038161003816119879119909 minus 1198791199101003816100381610038161003816 (|119909 minus 119879119909| +

1003816100381610038161003816119910 minus 1198791199101003816100381610038161003816 + 1)

= (2 minus radic3) (10038161003816100381610038161199101003816100381610038161003816 + 1) = (2 minus

radic3) (119910 + 1)

le (2 minus radic3 minus 119910)2

minus (2 minus radic3) (2 minus radic3 minus 119910)

= (1003816100381610038161003816119909 minus 119879119910

1003816100381610038161003816 +1003816100381610038161003816119910 minus 119879119909

1003816100381610038161003816)1003816100381610038161003816119909 minus 119910

1003816100381610038161003816

(18)

Thus

1003816100381610038161003816119879119909 minus 1198791199101003816100381610038161003816 le (

1003816100381610038161003816119909 minus 1198791199101003816100381610038161003816 +1003816100381610038161003816119910 minus 119879119909

1003816100381610038161003816

|119909 minus 119879119909| +1003816100381610038161003816119910 minus 119879119910

1003816100381610038161003816 + 1)1003816100381610038161003816119909 minus 119910

1003816100381610038161003816 (19)

Similar argument holds for the other conditions

Remark 4 Note that in (2) the ratio

119889 (119909 119879119910) + 119889 (119910 119879119909)

119889 (119909 119879119909) + 119889 (119910 119879119910) + 1(20)

might be greater or less than 1 and has not introduced anupper bound Note that if for every 119909 119910 isin 119883 119889(119909 119910) lt 12then we have

119889 (119909 119879119910) + 119889 (119910 119879119909)

le 2119889 (119909 119910) + 119889 (119909 119879119909) + 119889 (119910 119879119910)

lt 119889 (119909 119879119909) + 119889 (119910 119879119910) + 1

(21)

It means that

119889 (119909 119879119910) + 119889 (119910 119879119909)

119889 (119909 119879119909) + 119889 (119910 119879119910) + 1lt 1 (22)

and thus Theorem 1 is a special case of Banach contractionprinciple Therefore when (119883 119889) is a complete metric spacesuch that for all 119909 119910 isin 119883 119889(119909 119910) ge 12 Theorem 1 isvaluable because (20) might be greater than 1 Example 2shows this note precisely

The following is the second in our main results

Theorem 5 Let (119883 119889) be a complete metric space and let 119879 bea multivalued mapping from119883 into 119875

119888119897119887119889(119883) Let 119879 satisfy the

following

119867(119879119909 119879119910) le (119863 (119909 119879119910) + 119863 (119910 119879119909)

120575 (119909 119879119909) + 120575 (119910 119879119910) + 1)119889 (119909 119910) (23)

for all 119909 119910 isin 119883 Then 119879 has a fixed point isin 119883

Proof Let 1199090isin 119883 and 119909

1isin 1198791199090 For each 0 lt ℎ

1lt 1 one can

choose 1199092isin 1198791199091such that

119889 (1199091 1199092) lt 119867 (119879119909

0 1198791199091) + (1 minus

1

ℎ1

)119867(1198791199090 1198791199091)

=1

ℎ1

119867(1198791199090 1198791199091)

(24)

For each 0 lt ℎ119899lt 1 we can choose 119909

119899+1isin 119879119909119899such that

119889 (119909119899 119909119899+1) lt 119867 (119879119909

119899minus1 119879119909119899) + (1 minus

1

ℎ119899

)119867(1198791199090 1198791199091)

=1

ℎ119899

119867(1198791199090 1198791199091)

(25)

Specifically if

ℎ119899= radic

119889 (119909119899minus1+ 119909119899+1)

119889 (119909119899minus1+ 119909119899) + 119889 (119909

119899+ 119909119899+1) + 1

= radic120573119899

(26)

4 Abstract and Applied Analysis

then

119889 (119909119899 119909119899+1) le radic120573

119899119889 (119909119899minus1 119909119899) le 120573119899119889 (119909119899minus1 119909119899) (27)

Therefore119889 (119909119899+1 119909119899) le 120573119899119889 (119909119899 119909119899minus1)

le (120573119899120573119899minus1) 119889 (119909119899minus1 119909119899minus2)

le (120573119899120573119899minus1sdot sdot sdot 1205731) 119889 (1199091 1199090)

(28)

It can easily be seen that

lim119899rarrinfin

(12057311205732sdot sdot sdot 120573119899) = 0 (29)

Thus it is easily verified that

lim119899rarrinfin

119889 (119909119899+1 119909119899) = 0 (30)

Now for all119898 119899 isin N we have119889 (119909119898 119909119899) le 119889 (119909

119899 119909119899+1) + 119889 (119909

119899 119909119899+1) + sdot sdot sdot + 119889 (119909

119898minus1 119909119898)

le [(120573119899120573119899minus1sdot sdot sdot 1205731) + (120573

119899+1120573119899sdot sdot sdot 1205731)

+ sdot sdot sdot + (120573119898minus1120573119898minus2

sdot sdot sdot 1205731)] 119889 (119909

1 1199090)

=

119898minus1

sum

119896=119899

(120573119896120573119896minus1sdot sdot sdot 1205731) 119889 (1199091 1199090)

(31)

Suppose that 119886119896= (120573119896120573119896minus1sdot sdot sdot 1205731) Since

lim119896rarrinfin

119886119896+1

119886119896

= 0 (32)

suminfin

119896=1119886119896lt infin It means that

119898minus1

sum

119896=119899

(120573119896120573119896minus1sdot sdot sdot 1205731) 997888rarr 0 (33)

as 119898 119899 rarr infin In other words 119909119899 is a Cauchy sequence

and so converges to isin 119883 We claim that is a fixed pointConsider

119863 ( 119879) le 119889 ( 119909119899+1) + 119863 (119909

119899+1 119879)

le 119867 (119879119909119899 119879) + 119889 ( 119909

119899+1)

le (119863 ( 119879119909

119899) + 119863 (119909

119899 119879)

120575 ( 119879) + 120575 (119909119899 119879119909119899) + 1

)

times 119889 (119909119899 ) + 119889 ( 119909

119899+1)

le [119863 ( 119909119899+1) + 119863 (119909

119899 119879)]

times 119889 (119909119899 ) + 119889 ( 119909

119899+1)

(34)

On taking limit on both sides of (31) we have 119863( 119879) = 0It means that isin 119879

Remark 6 Note that Theorem 5 is a generalization ofTheorem 1 because by taking 119865119909 = 119879119909 and applyingTheorem 5 for 119865 we obtainTheorem 1

Conflict of Interests

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

References

[1] S Leader ldquoEquivalent Cauchy sequences and contractive fixedpoints in metric spacesrdquo Studia Mathematica vol 76 no 1 pp63ndash67 1983

[2] I A Rus ldquoPicard operators and applicationsrdquo Scientiae Mathe-maticae vol 58 no 1 pp 191ndash219 2003

[3] P V Subrahmanyam ldquoRemarks on some fixed point theoremsrelated to Banachrsquos contraction principlerdquo Journal of Mathemat-ical and Physical Sciences vol 8 pp 445ndash457 1974

[4] I A Rus A S Muresan and V Muresan ldquoWeakly Picardoperators on a set with two metricsrdquo Fixed Point Theory vol6 no 2 pp 323ndash331 2005

[5] J Caristi ldquoFixed point theorems for mappings satisfyinginwardness conditionsrdquo Transactions of the American Mathe-matical Society vol 215 pp 241ndash251 1976

[6] J Caristi and W A Kirk ldquoGeometric fixed point theory andinwardness conditionsrdquo in The Geometry of Metric and LinearSpaces vol 490 of Lecture Notes in Mathematics pp 74ndash83Springer Berlin Germany 1975

[7] S Banach ldquoSur les operations dans les ensembles abstraits etleur application aux equations integralesrdquo Fundamenta Mathe-maticae vol 3 pp 133ndash181 1922

[8] L B Ciric ldquoA generalization of Banachrsquos contraction principlerdquoProceedings of the American Mathematical Society vol 45 pp267ndash273 1974

[9] R Kannan ldquoSome results on fixed pointsmdashIIrdquo The AmericanMathematical Monthly vol 76 no 4 pp 405ndash408 1969

[10] W A Kirk ldquoFixed points of asymptotic contractionsrdquo Journalof Mathematical Analysis and Applications vol 277 no 2 pp645ndash650 2003

[11] J Matkowski ldquoIntegrable solutions of functional equationsrdquoDissertationes Mathematicae vol 127 pp 1ndash68 1975

[12] A Meir and E Keeler ldquoA theorem on contraction mappingsrdquoJournal of Mathematical Analysis and Applications vol 28 pp326ndash329 1969

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

[14] T Suzuki ldquoSeveral fixed point theorems concerning 120591-distancerdquoFixed Point Theory and Applications vol 2004 no 3 pp 195ndash209 2004

[15] T Suzuki ldquoA sufficient and necessary condition for the conver-gence of the sequence of successive approximations to a uniquefixed pointrdquo Proceedings of the American Mathematical Societyvol 136 no 11 pp 4089ndash4093 2008

[16] J S Bae ldquoFixed point theorems for weakly contractivemultival-ued mapsrdquo Journal of Mathematical Analysis and Applicationsvol 284 no 2 pp 690ndash697 2003

[17] J S Bae E W Cho and S H Yeom ldquoA generalization ofthe Caristi-Kirk fixed point theorem and its applications tomapping theoremsrdquo Journal of the KoreanMathematical Societyvol 31 no 1 pp 29ndash48 1994

[18] W A Kirk ldquoContraction mappings and extensionsrdquo in Hand-book of Metric Fixed PointTheory W A Kirk and B Sims Edspp 1ndash34 Kluwer Academic Dordrecht The Netherlands 2001

Abstract and Applied Analysis 5

[19] H AydiW ShatanawiM Postolache ZMustafa andN TahatldquoTheorems for Boyd-Wong-type contractions in orderedmetricspacesrdquo Abstract and Applied Analysis vol 2012 Article ID359054 14 pages 2012

[20] S Chandok andM Postolache ldquoFixed point theorem forweaklyChatterjea-type cyclic contractionsrdquo Fixed Point Theory andApplications vol 2013 no 28 2013

[21] S Chandok Z Mustafa andM Postolache ldquoCoupled commonfixed point results for mixed g-monotone mapps in partiallyordered G-metric spacesrdquo University Politehnica of BucharestScientific Bulletin A vol 75 no 4 pp 13ndash26 2013

[22] Y Chen ldquoStability of positive fixed points of nonlinear opera-torsrdquo Positivity vol 6 no 1 pp 47ndash57 2002

[23] B S Choudhury N Metiya and M Postolache ldquoA generalizedweak contraction principle with applications to coupled coin-cidence point problemsrdquo Fixed Point Theory and Applicationsvol 2013 article 152 2013

[24] R H Haghi M Postolache and S Rezapour ldquoOn T-stabilityof the Picard iteration for generalized 120593-contractionmappingsrdquoAbstract and Applied Analysis vol 2012 Article ID 658971 7pages 2012

[25] D Ilic and V Rakocevic ldquoCommon fixed points for mapson cone metric spacerdquo Journal of Mathematical Analysis andApplications vol 341 no 2 pp 876ndash882 2008

[26] J Jachymski ldquoThe contraction principle for mappings ona metric space with a graphrdquo Proceedings of the AmericanMathematical Society vol 136 no 4 pp 1359ndash1373 2008

[27] V Lakshmikantham and L Ciric ldquoCoupled fixed point the-orems for nonlinear contractions in partially ordered metricspacesrdquoNonlinearAnalysisTheoryMethodsampApplications vol70 no 12 pp 4341ndash4349 2009

[28] J J Nieto R L Pouso and R Rodriguez-Lopez ldquoFixedpoint theorems in ordered abstract spacesrdquo Proceedings of theAmerican Mathematical Society vol 135 no 8 pp 2505ndash25172007

[29] J J Nieto and R Rodriguez-Lopez ldquoContractive mappingtheorems in partially ordered sets and applications to ordinarydifferential equationsrdquo Order vol 22 no 3 pp 223ndash239 2005

[30] J J Nieto and R Rodrıguez-Lopez ldquoExistence and uniquenessof fixed point in partially ordered sets and applications toordinary differential equationsrdquo Acta Mathematica Sinica vol23 no 12 pp 2205ndash2212 2007

[31] D OrsquoRegan and A Petrusel ldquoFixed point theorems for gen-eralized contractions in ordered metric spacesrdquo Journal ofMathematical Analysis andApplications vol 341 no 2 pp 1241ndash1252 2008

[32] H K Pathak and N Shahzad ldquoFixed points for generalizedcontractions and applications to control theoryrdquo NonlinearAnalysisTheoryMethodsampApplications vol 68 no 8 pp 2181ndash2193 2008

[33] W Shatanawi and M Postolache ldquoCommon fixed point theo-rems for dominating andweak annihilatormappings in orderedmetric spacesrdquo Fixed Point Theory and Applications vol 2013article 271 2013

[34] W Shatanawi and M Postolache ldquoCommon fixed point resultsfor mappings under nonlinear contraction of cyclic form inorderedmetric spacesrdquo Fixed PointTheory andApplications vol2013 no 60 2013

[35] Y Wu ldquoNew fixed point theorems and applications of mixedmonotone operatorrdquo Journal of Mathematical Analysis andApplications vol 341 no 2 pp 883ndash893 2008

[36] J Nadler ldquoMulti-valued contraction mappingsrdquo Pacific Journalof Mathematics vol 30 pp 475ndash488 1969

[37] A Amini-Harandi ldquoEndpoints of set-valued contractions inmetric spacesrdquo Nonlinear Analysis Theory Methods amp Applica-tions vol 72 no 1 pp 132ndash134 2010

[38] A Amini-Harandi ldquoFixed point theory for set-valued quasi-contraction maps in metric spacesrdquo Applied Mathematics Let-ters vol 24 no 11 pp 1791ndash1794 2011

[39] L B Ciric and J S Ume ldquoMulti-valued non-self-mappings onconvex metric spacesrdquo Nonlinear Analysis Theory Methods ampApplications vol 60 no 6 pp 1053ndash1063 2005

[40] M Fakhar ldquoEndpoints of set-valued asymptotic contractions inmetric spacesrdquo Applied Mathematics Letters vol 24 no 4 pp428ndash431 2011

[41] N Hussain A Amini-Harandi and Y J Cho ldquoApproximateendpoints for set-valued contractions in metric spacesrdquo FixedPoint Theory and Applications vol 2010 Article ID 614867 13pages 2010

[42] Z Kadelburg and S Radenovic ldquoSome results on set-valuedcontractions in abstract metric spacesrdquo Computers amp Mathe-matics with Applications vol 62 no 1 pp 342ndash350 2011

[43] S Moradi and F Khojasteh ldquoEndpoints of multi-valued gener-alized weak contraction mappingsrdquo Nonlinear Analysis TheoryMethods amp Applications vol 74 no 6 pp 2170ndash2174 2011

[44] F Khojasteh and Rakocevic V ldquoSome new common fixedpoint results for generalized contractive multi-valued non-self-mappingsrdquo Applied Mathematics Letters vol 25 no 3 pp 287ndash293 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Abstract and Applied Analysis

then

119889 (119909119899 119909119899+1) le radic120573

119899119889 (119909119899minus1 119909119899) le 120573119899119889 (119909119899minus1 119909119899) (27)

Therefore119889 (119909119899+1 119909119899) le 120573119899119889 (119909119899 119909119899minus1)

le (120573119899120573119899minus1) 119889 (119909119899minus1 119909119899minus2)

le (120573119899120573119899minus1sdot sdot sdot 1205731) 119889 (1199091 1199090)

(28)

It can easily be seen that

lim119899rarrinfin

(12057311205732sdot sdot sdot 120573119899) = 0 (29)

Thus it is easily verified that

lim119899rarrinfin

119889 (119909119899+1 119909119899) = 0 (30)

Now for all119898 119899 isin N we have119889 (119909119898 119909119899) le 119889 (119909

119899 119909119899+1) + 119889 (119909

119899 119909119899+1) + sdot sdot sdot + 119889 (119909

119898minus1 119909119898)

le [(120573119899120573119899minus1sdot sdot sdot 1205731) + (120573

119899+1120573119899sdot sdot sdot 1205731)

+ sdot sdot sdot + (120573119898minus1120573119898minus2

sdot sdot sdot 1205731)] 119889 (119909

1 1199090)

=

119898minus1

sum

119896=119899

(120573119896120573119896minus1sdot sdot sdot 1205731) 119889 (1199091 1199090)

(31)

Suppose that 119886119896= (120573119896120573119896minus1sdot sdot sdot 1205731) Since

lim119896rarrinfin

119886119896+1

119886119896

= 0 (32)

suminfin

119896=1119886119896lt infin It means that

119898minus1

sum

119896=119899

(120573119896120573119896minus1sdot sdot sdot 1205731) 997888rarr 0 (33)

as 119898 119899 rarr infin In other words 119909119899 is a Cauchy sequence

and so converges to isin 119883 We claim that is a fixed pointConsider

119863 ( 119879) le 119889 ( 119909119899+1) + 119863 (119909

119899+1 119879)

le 119867 (119879119909119899 119879) + 119889 ( 119909

119899+1)

le (119863 ( 119879119909

119899) + 119863 (119909

119899 119879)

120575 ( 119879) + 120575 (119909119899 119879119909119899) + 1

)

times 119889 (119909119899 ) + 119889 ( 119909

119899+1)

le [119863 ( 119909119899+1) + 119863 (119909

119899 119879)]

times 119889 (119909119899 ) + 119889 ( 119909

119899+1)

(34)

On taking limit on both sides of (31) we have 119863( 119879) = 0It means that isin 119879

Remark 6 Note that Theorem 5 is a generalization ofTheorem 1 because by taking 119865119909 = 119879119909 and applyingTheorem 5 for 119865 we obtainTheorem 1

Conflict of Interests

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

References

[1] S Leader ldquoEquivalent Cauchy sequences and contractive fixedpoints in metric spacesrdquo Studia Mathematica vol 76 no 1 pp63ndash67 1983

[2] I A Rus ldquoPicard operators and applicationsrdquo Scientiae Mathe-maticae vol 58 no 1 pp 191ndash219 2003

[3] P V Subrahmanyam ldquoRemarks on some fixed point theoremsrelated to Banachrsquos contraction principlerdquo Journal of Mathemat-ical and Physical Sciences vol 8 pp 445ndash457 1974

[4] I A Rus A S Muresan and V Muresan ldquoWeakly Picardoperators on a set with two metricsrdquo Fixed Point Theory vol6 no 2 pp 323ndash331 2005

[5] J Caristi ldquoFixed point theorems for mappings satisfyinginwardness conditionsrdquo Transactions of the American Mathe-matical Society vol 215 pp 241ndash251 1976

[6] J Caristi and W A Kirk ldquoGeometric fixed point theory andinwardness conditionsrdquo in The Geometry of Metric and LinearSpaces vol 490 of Lecture Notes in Mathematics pp 74ndash83Springer Berlin Germany 1975

[7] S Banach ldquoSur les operations dans les ensembles abstraits etleur application aux equations integralesrdquo Fundamenta Mathe-maticae vol 3 pp 133ndash181 1922

[8] L B Ciric ldquoA generalization of Banachrsquos contraction principlerdquoProceedings of the American Mathematical Society vol 45 pp267ndash273 1974

[9] R Kannan ldquoSome results on fixed pointsmdashIIrdquo The AmericanMathematical Monthly vol 76 no 4 pp 405ndash408 1969

[10] W A Kirk ldquoFixed points of asymptotic contractionsrdquo Journalof Mathematical Analysis and Applications vol 277 no 2 pp645ndash650 2003

[11] J Matkowski ldquoIntegrable solutions of functional equationsrdquoDissertationes Mathematicae vol 127 pp 1ndash68 1975

[12] A Meir and E Keeler ldquoA theorem on contraction mappingsrdquoJournal of Mathematical Analysis and Applications vol 28 pp326ndash329 1969

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

[14] T Suzuki ldquoSeveral fixed point theorems concerning 120591-distancerdquoFixed Point Theory and Applications vol 2004 no 3 pp 195ndash209 2004

[15] T Suzuki ldquoA sufficient and necessary condition for the conver-gence of the sequence of successive approximations to a uniquefixed pointrdquo Proceedings of the American Mathematical Societyvol 136 no 11 pp 4089ndash4093 2008

[16] J S Bae ldquoFixed point theorems for weakly contractivemultival-ued mapsrdquo Journal of Mathematical Analysis and Applicationsvol 284 no 2 pp 690ndash697 2003

[17] J S Bae E W Cho and S H Yeom ldquoA generalization ofthe Caristi-Kirk fixed point theorem and its applications tomapping theoremsrdquo Journal of the KoreanMathematical Societyvol 31 no 1 pp 29ndash48 1994

[18] W A Kirk ldquoContraction mappings and extensionsrdquo in Hand-book of Metric Fixed PointTheory W A Kirk and B Sims Edspp 1ndash34 Kluwer Academic Dordrecht The Netherlands 2001

Abstract and Applied Analysis 5

[19] H AydiW ShatanawiM Postolache ZMustafa andN TahatldquoTheorems for Boyd-Wong-type contractions in orderedmetricspacesrdquo Abstract and Applied Analysis vol 2012 Article ID359054 14 pages 2012

[20] S Chandok andM Postolache ldquoFixed point theorem forweaklyChatterjea-type cyclic contractionsrdquo Fixed Point Theory andApplications vol 2013 no 28 2013

[21] S Chandok Z Mustafa andM Postolache ldquoCoupled commonfixed point results for mixed g-monotone mapps in partiallyordered G-metric spacesrdquo University Politehnica of BucharestScientific Bulletin A vol 75 no 4 pp 13ndash26 2013

[22] Y Chen ldquoStability of positive fixed points of nonlinear opera-torsrdquo Positivity vol 6 no 1 pp 47ndash57 2002

[23] B S Choudhury N Metiya and M Postolache ldquoA generalizedweak contraction principle with applications to coupled coin-cidence point problemsrdquo Fixed Point Theory and Applicationsvol 2013 article 152 2013

[24] R H Haghi M Postolache and S Rezapour ldquoOn T-stabilityof the Picard iteration for generalized 120593-contractionmappingsrdquoAbstract and Applied Analysis vol 2012 Article ID 658971 7pages 2012

[25] D Ilic and V Rakocevic ldquoCommon fixed points for mapson cone metric spacerdquo Journal of Mathematical Analysis andApplications vol 341 no 2 pp 876ndash882 2008

[26] J Jachymski ldquoThe contraction principle for mappings ona metric space with a graphrdquo Proceedings of the AmericanMathematical Society vol 136 no 4 pp 1359ndash1373 2008

[27] V Lakshmikantham and L Ciric ldquoCoupled fixed point the-orems for nonlinear contractions in partially ordered metricspacesrdquoNonlinearAnalysisTheoryMethodsampApplications vol70 no 12 pp 4341ndash4349 2009

[28] J J Nieto R L Pouso and R Rodriguez-Lopez ldquoFixedpoint theorems in ordered abstract spacesrdquo Proceedings of theAmerican Mathematical Society vol 135 no 8 pp 2505ndash25172007

[29] J J Nieto and R Rodriguez-Lopez ldquoContractive mappingtheorems in partially ordered sets and applications to ordinarydifferential equationsrdquo Order vol 22 no 3 pp 223ndash239 2005

[30] J J Nieto and R Rodrıguez-Lopez ldquoExistence and uniquenessof fixed point in partially ordered sets and applications toordinary differential equationsrdquo Acta Mathematica Sinica vol23 no 12 pp 2205ndash2212 2007

[31] D OrsquoRegan and A Petrusel ldquoFixed point theorems for gen-eralized contractions in ordered metric spacesrdquo Journal ofMathematical Analysis andApplications vol 341 no 2 pp 1241ndash1252 2008

[32] H K Pathak and N Shahzad ldquoFixed points for generalizedcontractions and applications to control theoryrdquo NonlinearAnalysisTheoryMethodsampApplications vol 68 no 8 pp 2181ndash2193 2008

[33] W Shatanawi and M Postolache ldquoCommon fixed point theo-rems for dominating andweak annihilatormappings in orderedmetric spacesrdquo Fixed Point Theory and Applications vol 2013article 271 2013

[34] W Shatanawi and M Postolache ldquoCommon fixed point resultsfor mappings under nonlinear contraction of cyclic form inorderedmetric spacesrdquo Fixed PointTheory andApplications vol2013 no 60 2013

[35] Y Wu ldquoNew fixed point theorems and applications of mixedmonotone operatorrdquo Journal of Mathematical Analysis andApplications vol 341 no 2 pp 883ndash893 2008

[36] J Nadler ldquoMulti-valued contraction mappingsrdquo Pacific Journalof Mathematics vol 30 pp 475ndash488 1969

[37] A Amini-Harandi ldquoEndpoints of set-valued contractions inmetric spacesrdquo Nonlinear Analysis Theory Methods amp Applica-tions vol 72 no 1 pp 132ndash134 2010

[38] A Amini-Harandi ldquoFixed point theory for set-valued quasi-contraction maps in metric spacesrdquo Applied Mathematics Let-ters vol 24 no 11 pp 1791ndash1794 2011

[39] L B Ciric and J S Ume ldquoMulti-valued non-self-mappings onconvex metric spacesrdquo Nonlinear Analysis Theory Methods ampApplications vol 60 no 6 pp 1053ndash1063 2005

[40] M Fakhar ldquoEndpoints of set-valued asymptotic contractions inmetric spacesrdquo Applied Mathematics Letters vol 24 no 4 pp428ndash431 2011

[41] N Hussain A Amini-Harandi and Y J Cho ldquoApproximateendpoints for set-valued contractions in metric spacesrdquo FixedPoint Theory and Applications vol 2010 Article ID 614867 13pages 2010

[42] Z Kadelburg and S Radenovic ldquoSome results on set-valuedcontractions in abstract metric spacesrdquo Computers amp Mathe-matics with Applications vol 62 no 1 pp 342ndash350 2011

[43] S Moradi and F Khojasteh ldquoEndpoints of multi-valued gener-alized weak contraction mappingsrdquo Nonlinear Analysis TheoryMethods amp Applications vol 74 no 6 pp 2170ndash2174 2011

[44] F Khojasteh and Rakocevic V ldquoSome new common fixedpoint results for generalized contractive multi-valued non-self-mappingsrdquo Applied Mathematics Letters vol 25 no 3 pp 287ndash293 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 5

[19] H AydiW ShatanawiM Postolache ZMustafa andN TahatldquoTheorems for Boyd-Wong-type contractions in orderedmetricspacesrdquo Abstract and Applied Analysis vol 2012 Article ID359054 14 pages 2012

[20] S Chandok andM Postolache ldquoFixed point theorem forweaklyChatterjea-type cyclic contractionsrdquo Fixed Point Theory andApplications vol 2013 no 28 2013

[21] S Chandok Z Mustafa andM Postolache ldquoCoupled commonfixed point results for mixed g-monotone mapps in partiallyordered G-metric spacesrdquo University Politehnica of BucharestScientific Bulletin A vol 75 no 4 pp 13ndash26 2013

[22] Y Chen ldquoStability of positive fixed points of nonlinear opera-torsrdquo Positivity vol 6 no 1 pp 47ndash57 2002

[23] B S Choudhury N Metiya and M Postolache ldquoA generalizedweak contraction principle with applications to coupled coin-cidence point problemsrdquo Fixed Point Theory and Applicationsvol 2013 article 152 2013

[24] R H Haghi M Postolache and S Rezapour ldquoOn T-stabilityof the Picard iteration for generalized 120593-contractionmappingsrdquoAbstract and Applied Analysis vol 2012 Article ID 658971 7pages 2012

[25] D Ilic and V Rakocevic ldquoCommon fixed points for mapson cone metric spacerdquo Journal of Mathematical Analysis andApplications vol 341 no 2 pp 876ndash882 2008

[26] J Jachymski ldquoThe contraction principle for mappings ona metric space with a graphrdquo Proceedings of the AmericanMathematical Society vol 136 no 4 pp 1359ndash1373 2008

[27] V Lakshmikantham and L Ciric ldquoCoupled fixed point the-orems for nonlinear contractions in partially ordered metricspacesrdquoNonlinearAnalysisTheoryMethodsampApplications vol70 no 12 pp 4341ndash4349 2009

[28] J J Nieto R L Pouso and R Rodriguez-Lopez ldquoFixedpoint theorems in ordered abstract spacesrdquo Proceedings of theAmerican Mathematical Society vol 135 no 8 pp 2505ndash25172007

[29] J J Nieto and R Rodriguez-Lopez ldquoContractive mappingtheorems in partially ordered sets and applications to ordinarydifferential equationsrdquo Order vol 22 no 3 pp 223ndash239 2005

[30] J J Nieto and R Rodrıguez-Lopez ldquoExistence and uniquenessof fixed point in partially ordered sets and applications toordinary differential equationsrdquo Acta Mathematica Sinica vol23 no 12 pp 2205ndash2212 2007

[31] D OrsquoRegan and A Petrusel ldquoFixed point theorems for gen-eralized contractions in ordered metric spacesrdquo Journal ofMathematical Analysis andApplications vol 341 no 2 pp 1241ndash1252 2008

[32] H K Pathak and N Shahzad ldquoFixed points for generalizedcontractions and applications to control theoryrdquo NonlinearAnalysisTheoryMethodsampApplications vol 68 no 8 pp 2181ndash2193 2008

[33] W Shatanawi and M Postolache ldquoCommon fixed point theo-rems for dominating andweak annihilatormappings in orderedmetric spacesrdquo Fixed Point Theory and Applications vol 2013article 271 2013

[34] W Shatanawi and M Postolache ldquoCommon fixed point resultsfor mappings under nonlinear contraction of cyclic form inorderedmetric spacesrdquo Fixed PointTheory andApplications vol2013 no 60 2013

[35] Y Wu ldquoNew fixed point theorems and applications of mixedmonotone operatorrdquo Journal of Mathematical Analysis andApplications vol 341 no 2 pp 883ndash893 2008

[36] J Nadler ldquoMulti-valued contraction mappingsrdquo Pacific Journalof Mathematics vol 30 pp 475ndash488 1969

[37] A Amini-Harandi ldquoEndpoints of set-valued contractions inmetric spacesrdquo Nonlinear Analysis Theory Methods amp Applica-tions vol 72 no 1 pp 132ndash134 2010

[38] A Amini-Harandi ldquoFixed point theory for set-valued quasi-contraction maps in metric spacesrdquo Applied Mathematics Let-ters vol 24 no 11 pp 1791ndash1794 2011

[39] L B Ciric and J S Ume ldquoMulti-valued non-self-mappings onconvex metric spacesrdquo Nonlinear Analysis Theory Methods ampApplications vol 60 no 6 pp 1053ndash1063 2005

[40] M Fakhar ldquoEndpoints of set-valued asymptotic contractions inmetric spacesrdquo Applied Mathematics Letters vol 24 no 4 pp428ndash431 2011

[41] N Hussain A Amini-Harandi and Y J Cho ldquoApproximateendpoints for set-valued contractions in metric spacesrdquo FixedPoint Theory and Applications vol 2010 Article ID 614867 13pages 2010

[42] Z Kadelburg and S Radenovic ldquoSome results on set-valuedcontractions in abstract metric spacesrdquo Computers amp Mathe-matics with Applications vol 62 no 1 pp 342ndash350 2011

[43] S Moradi and F Khojasteh ldquoEndpoints of multi-valued gener-alized weak contraction mappingsrdquo Nonlinear Analysis TheoryMethods amp Applications vol 74 no 6 pp 2170ndash2174 2011

[44] F Khojasteh and Rakocevic V ldquoSome new common fixedpoint results for generalized contractive multi-valued non-self-mappingsrdquo Applied Mathematics Letters vol 25 no 3 pp 287ndash293 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of