research article common fixed point of two -weakly...

Research ArticleCommon Fixed Point of Two 119877-Weakly Commuting Mappingsin 119887-Metric Spaces

Poom Kumam12 Wutiphol Sintunavarat3 Shaban Sedghi4 and Nabi Shobkolaei5

1Department of Mathematics Faculty of Science King Mongkutrsquos University of Technology Thonburi (KMUTT)126 Pracha Uthit Road Bang Mod Thrung Khru Bangkok 10140 Thailand2Centre of Excellence in Mathematics CHE Si Ayutthaya Road Bangkok 10400 Thailand3Department of Mathematics and Statistics Faculty of Science and Technology Thammasat University Rangsit CenterPathumthani 12121 Thailand4Department of Mathematics Islamic Azad University Qaemshahr Branch Qaemshahr Iran5Department of Mathematics Islamic Azad University Babol Branch Babol Iran

Correspondence should be addressed to Wutiphol Sintunavarat wutipholmathstatscituacth

Received 21 June 2014 Accepted 9 December 2014

Academic Editor Calogero Vetro

Copyright copy 2015 Poom Kumam 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 prove some common fixed point results for two mappings satisfying generalized contractive condition in 119887-metric space Notethat 119887-metric of main results in this work are not necessarily continuous So our results extend and improve several previous worksWe also present one example that shows the applicability and usefulness of our results

1 Introduction

In 1998 Czerwik [1] introduced the concept of 119887-metric spaceSince then several papers deal with fixed point theory forsingle-valued and multivalued operators in 119887-metric spaces(see also [1ndash19]) For example Pacurar [16] proved resultson sequences of almost contractions and fixed points in 119887-metric spaces Also Hussain and Shah [11] obtained resultson KKM mappings in cone 119887-metric spaces FurthermoreKhamsi ([12 13]) also showed that each conemetric space hasa 119887-metric structure

The aim of this paper is to present some common fixedpoint results for two mappings under generalized contractivecondition in 119887-metric space where the 119887-metric is notnecessarily continuous Because many of the authors in theirworks have used the 119887-metric spaces in which the 119887-metricis continuous the techniques used in this paper can be usedformany of the results on the context of 119887-metric space Fromthis point of view the results obtained in this paper generalizeand extend several ones obtained earlier in a lot of papersconcerning 119887-metric space

2 Preliminaries

Throughout this paper we denote by N R+ and R the

sets of positive integers nonnegative real numbers and realnumbers respectively

Consistent with [1] and [18 page 264] the followingdefinition and results will be needed in the sequel

Definition 1 (see [1]) Let 119883 be a nonempty set and let 119887 ge 1

be a given real number A function 119889 119883 times 119883 rarr R+is a

119887-metric if and only if for all 119909 119910 119911 isin 119883 the followingconditions are satisfied

(b1) 119889(119909 119910) = 0 if and only if 119909 = 119910(b2) 119889(119909 119910) = 119889(119910 119909)(b3) 119889(119909 119911) le 119887[119889(119909 119910) + 119889(119910 119911)]

The pair (119883 119889) is called a 119887-metric space with coefficient 119887

It should be noted that the class of 119887-metric spaces iseffectively larger than that of metric spaces since a 119887-metricis a metric when 119887 = 1

Hindawi Publishing CorporationJournal of Function SpacesVolume 2015 Article ID 350840 5 pageshttpdxdoiorg1011552015350840

2 Journal of Function Spaces

We present an example which shows that a 119887-metric on119883need not be a metric on119883 (see also [18 page 264])

Example 2 Let (119883 119889) be a metric space and 120588(119909 119910) =

(119889(119909 119910))119901 where 119901 gt 1 is a real number We show that 120588

is a 119887-metric with 119887 = 2119901minus1Obviously conditions (b1) and (b2) in Definition 1 are

satisfied Now we show that condition (b3) holds for 120588It is easy to see that if 1 lt 119901 lt infin then the convexity of

the function 119891(119909) = 119909119901 where 119909 ge 0 implies

(119886 + 119888

2)

119901

le1

2(119886119901

+ 119888119901

) (1)

and hence

(119886 + 119888)119901

le 2119901minus1

(119886119901

+ 119888119901

) (2)

Therefore for each 119909 119910 119911 isin 119883 we obtain that

120588 (119909 119910) = (119889 (119909 119910))119901

le (119889 (119909 119911) + 119889 (119911 119910))119901

le 2119901minus1

((119889 (119909 119911))119901

+ (119889 (119911 119910))119901

)

= 2119901minus1

(120588 (119909 119911) + 120588 (119911 119910))

(3)

So condition (b3) in Definition 1 holds and then 120588 is a 119887-metric coefficient 119904 = 2119901minus1

It should be noted that in Example 2 if (119883 119889) is a metricspace then (119883 120588) is not necessarily a metric space (seeExample 3)

Example 3 For example if 119883 = R and the usual Euclideanmetric 119889 119883 times 119883 rarr R

+is defined by

119889 (119909 119910) =1003816100381610038161003816119909 minus 119910

1003816100381610038161003816 (4)

for all 119909 119910 isin 119883 then 120588(119909 119910) = |119909minus119910|2 is a 119887-metric onRwith119887 = 2 but is not a metric onR because the triangle inequalitydoes not hold

Before stating and proving our results we present somedefinition and proposition in 119887-metric space We recall firstthe notions of convergence closedness and completeness ina 119887-metric space

Definition 4 (see [6]) Let (119883 119889) be a 119887-metric space Then asequence 119909

119899 in119883 is called

(i) convergent if and only if there exists 119909 isin 119883 such that119889(119909119899 119909) rarr 0 as 119899 rarr +infin and in this case we write

lim119899rarrinfin

119909119899= 119909

(ii) Cauchy if and only if 119889(119909119899 119909119898) rarr 0 as 119899119898 rarr +infin

Proposition 5 (Remark 21 in [6]) In a 119887-metric space (119883 119889)the following assertions hold

(i) a convergent sequence has a unique limit

(ii) each convergent sequence is Cauchy(iii) in general a 119887-metric is not continuous

Definition 6 (see [6]) The 119887-metric space (119883 119889) is completeif every Cauchy sequence in119883 converges

It should be noted that in general a 119887-metric function119889 for119887 gt 1 is not jointly continuous in all the two of its variables

Since in general a 119887-metric is not continuous we need thefollowing simple lemma about the 119887-convergent sequences

Lemma 7 (see [20]) Let (119883 119889) be a 119887-metric space withcoefficient 119887 ge 1 and suppose that 119909

119899 and 119910

119899 are 119887-

convergent to 119909 119910 respectively then one has

1

1198872119889 (119909 119910) le lim inf

119899rarrinfin

119889 (119909119899 119910119899) le lim sup119899rarrinfin

119889 (119909119899 119910119899)

le 1198872

119889 (119909 119910)

(5)

In particular if 119909 = 119910 then one has lim119899rarrinfin

119889(119909119899 119910119899) = 0

Moreover for each 119911 isin 119883 one has

1

119887119889 (119909 119911) le lim inf

119899rarrinfin

119889 (119909119899 119911) le lim sup

119899rarrinfin

119889 (119909119899 119911) le 119887119889 (119909 119911)

(6)

3 Common Fixed Point Results

Definition 8 Let 119891 and 119892 be mappings from a 119887-metric space(119883 119889) into itself The mappings 119891 and 119892 are said to be weaklycommuting if

119889 (119891119892119909 119892119891119909) le 119889 (119891119909 119892119909) (7)

for each 119909 in119883

Definition 9 Let 119891 and 119892 be mappings from a 119887-metric space(119883 119889) into itself The mappings 119891 and 119892 are said to be 119877-weakly commuting if there exists some positive real number119877 such that

119889 (119891119892119909 119892119891119909) le 119877119889 (119891119909 119892119909) (8)

for each 119909 in119883

Remark 10 Weak commutativity implies 119877-weak commu-tativity in 119887-metric space However 119877-weak commutativityimplies weak commutativity only when 119877 le 1

Example 11 Let 119883 = R and 119889 119883 times 119883 rarr R+defined as

follows

119889 (119909 119910) = (119909 minus 119910)2 (9)

for all 119909 119910 isin 119883 Then (119883 119889) is a 119887-metric space Define 119891119909 =2119909 minus 1 and 119892119909 = 1199092 Then

119889 (119891119892119909 119892119891119909) = 4 (119909 minus 1)4

= 4119889 (119891119909 119892119909) gt 119889 (119891119909 119892119909) (10)

Therefore for 119877 = 4 119891 and 119892 are 119877-weakly commuting But119891 and 119892 are not weakly commuting

Journal of Function Spaces 3

Theorem 12 Let (119883 119889) be a complete 119887-metric space and let119891 and 119892 be 119877-weakly commuting self-mappings on119883 satisfyingthe following conditions

(a) 119891(119883) sube 119892(119883)(b) 119891 or 119892 is continuous(c) 119889(119891119909 119891119910) le 120574((1119887

4)119889(119892119909 119892119910)) for all 119909 119910 isin 119883

where 120574 [0infin) rarr [0infin) is a continuous andnondecreasing function such that 120574(119886) lt 119886 for each119886 gt 0 and 120574(0) = 0

Then 119891 and 119892 have a unique common fixed point

Proof Let 1199090be an arbitrary point in119883 By (a) choose a point

1199091in119883 such that119891119909

0= 1198921199091 In general choose 119909

119899+1such that

119891119909119899= 119892119909119899+1

for all 119899 isin Ncup 0 Now we observe that for each119899 isin N we have

119889 (119891119909119899 119891119909119899+1

) le 120574 (1

1198874119889 (119892119909119899 119892119909119899+1

))

= 120574 (1

1198874119889 (119891119909119899minus1

119891119909119899))

le1

1198874119889 (119891119909119899minus1

119891119909119899)

le 119889 (119891119909119899minus1

119891119909119899)

(11)

This implies that

119888119899 = 119889 (119891119909

119899 119891119909119899+1

) (12)

is nonincreasing sequence in [0infin) Therefore it tends to alimit 119886 ge 0 Next we claim that 119886 = 0 Suppose that 119886 gt 0Making 119899 rarr infin in the inequality (11) we get

119886 le 120574 (1

1198874119886) lt

1

1198874119886 le 119886 (13)

which is a contradiction Hence 119886 = 0 that is

lim119899rarrinfin

119888119899= lim119899rarrinfin

119889 (119891119909119899 119891119909119899+1

) = 0 (14)

Now we prove that 119891119909119899 is a Cauchy sequence in 119891(119883)

Suppose that 119891119909119899 is not a Cauchy sequence in 119891(119883) For

convenience let 119910119899= 119891119909119899for 119899 = 1 2 3 Then there is an

120576 gt 0 such that for each integer 119896 there exist integers 119898(119896)and 119899(119896) with119898(119896) gt 119899(119896) ge 119896 such that

119889119896= 119889 (119910

119899(119896) 119910119898(119896)

) ge 120576 for 119896 = 1 2 (15)

We may assume that

119889 (119910119899(119896)

119910119898(119896)minus1

) lt 120576 (16)

by choosing 119898(119896) as the smallest number exceeding 119899(119896) forwhich (15) holds Using (11) we have

120576 le 119889119896le 119887 [119889 (119910

119899(119896) 119910119898(119896)minus1

) + 119889 (119910119898(119896)minus1

119910119898(119896)

)]

lt 119887120576 + 119887119888119898(119896)minus1

le 119887120576 + 119887119888119896

(17)

Hence 120576 le lim119896rarrinfin

119889119896le 119887120576 Also notice

119889119896= 119889 (119910

119899(119896) 119910119898(119896)

)

le 119887119889 (119910119899(119896)

119910119899(119896)+1

)

+ 1198872

[119889 (119910119899(119896)+1

119910119898(119896)+1

) + 119889 (119910119898(119896)+1

119910119898(119896)

)]

le 119887119888119896+ 1198872

120574 (1

1198874119889 (119910119899(119896)

119910119898(119896)

)) + 1198872

119888119896

= 119887119888119896+ 1198872

120574 (1

1198874119889119896) + 1198872

119888119896

(18)

for each 119896 isin N Thus as 119896 rarr infin in the above inequality wehave

120576 le lim119896rarrinfin

119889119896le 119887 lim119896rarrinfin

119888119896+ 1198872

120574 (1

1198874lim119896rarrinfin

119889119896) + 1198872 lim119896rarrinfin

119888119896

(19)

and thus

120576 le 1198872

120574 (1

1198873120576) lt

1

119887120576 (20)

which is a contradiction Thus 119891119909119899 is Cauchy sequence

in 119883 and by the completeness of 119883 119891119909119899 converges to 119911

in 119883 Also 119892119909119899 converges to 119911 in 119883 Let us suppose that

the mapping 119891 is continuous Then lim119899rarrinfin

119891119891119909119899= 119891119911 and

lim119899rarrinfin

119891119892119909119899= 119891119911 Further we have that since 119891 and 119892 are

119877-weakly commuting

119889 (119891119892119909119899 119892119891119909119899) le 119877119889 (119891119909

119899 119892119909119899) (21)

for all 119899 isin N Taking the upper limit as 119899 rarr infin in the aboveinequality we get

1

1198872119889(119891119911 lim sup

119899rarrinfin

119892119891119909119899) le lim sup119899rarrinfin

119889 (119891119892119909119899 119892119891119909119899)

le 119877lim sup119899rarrinfin

119889 (119891119909119899 119892119909119899)

le 1198771198872

119889 (119911 119911) = 0

(22)

Similarly

1

1198872119889 (119891119911 lim inf

119899rarrinfin

119892119891119909119899) = 0 (23)

and hence we get lim119899rarrinfin

119892119891119909119899= 119891119911 We now prove that 119911 =

119891119911 Suppose that 119911 = 119891119911 and then 119889(119911 119891119911) gt 0 By (c) wehave

1

119887119889 (119891119911 119891119909


119889 (119891119891119909119899 119891119909119898)

le 120574(lim sup119899rarrinfin

1

1198874119889 (119892119891119909

119899 119892119909119898))

le 120574 (1

1198873119889 (119891119911 119892119909

119898))

(24)


for each119898 isin N On taking the upper limit as119898 rarr infin in theabove inequality we get

1

1198872119889 (119891119911 119911) le lim sup

119898rarrinfin

1

119887119889 (119891119911 119891119909

119898)


1

1198873119889 (119891119911 119892119909

119898))

le 120574 (1

1198872119889 (119891119911 119911)) lt

1

1198872119889 (119891119911 119911)

(25)

which is a contradiction Therefore 119911 = 119891119911 Since 119891(119883) sube119892(119883) we can find 119911

1in 119883 such that 119911 = 119891119911 = 119892119911

1 Now we

have

119889 (119891119891119909119899 1198911199111) le 120574 (

1

1198874119889 (119892119891119909

119899 1198921199111)) (26)

for all 119899 isin N Taking limit sup as 119899 rarr infin we get

1

119887119889 (119891119911 119891119911


119889 (119891119891119909119899 119891119911)


1

1198874119889 (119892119891119909

119899 1198921199111))

le 120574 (1

1198873119889 (119891119911 119892119911

1))

= 0

(27)

since 120574(0) = 0 which implies that 119891119911 = 1198911199111 that is 119911 = 119891119911 =

1198911199111= 1198921199111 Also

119889 (119891119911 119892119911) = 119889 (1198911198921199111 1198921198911199111) le 119877119889 (119891119911

1 1198921199111) = 0 (28)

which again implies that 119891119911 = 119892119911 Thus 119911 is a common fixedpoint of 119891 and 119892

Now to prove uniqueness let if possible 1199111015840 = 119911 be anothercommon fixed point of 119891 and 119892 Then 119889(119911 1199111015840) gt 0 and so

119889 (119911 1199111015840

) = 119889 (119891119911 1198911199111015840

) le 120574 (1

1198874119889 (119892119911 119892119911

1015840

))

= 120574 (1

1198874119889 (119911 119911

1015840

)) lt1

1198874119889 (119911 119911

1015840

) le 119889 (119911 1199111015840

)

(29)

which is a contradiction Therefore 119911 = 1199111015840 that is 119911 is a

unique common fixed point of 119891 and 119892 This completes theproof

Now we give an example to support Theorem 12

Example 13 Let119883 = R and 119889 119883 times 119883 rarr R+defined by

119889 (119909 119910) = (119909 minus 119910)2 (30)

for all 119909 119910 isin 119883 Then (119883 119889) is a 119887-metric space for 119887 = 2Define119891119909 = 1 and 119892119909 = 2119909minus1 on119883 It is evident that119891(119883) sube119892(119883) and 119891 is continuous Now we observe that

119889 (119891119909 119891119910) le 120574 (1

16119889 (119892119909 119892119910)) (31)

for all 119909 119910 in119883 and 120574 (0infin) rarr (0infin) defined by 120574(119905) = 119896119905for 0 lt 119896 lt 1 Moreover it is easy to see that 119891 and 119892 are 119877-weakly commutingThus all the conditions ofTheorem 12 aresatisfied and 1 is a common fixed point of 119891 and 119892

Corollary 14 Let (119883 119889) be a complete 119887-metric space and let119891 be a self-mapping on 119883 satisfying the following condition

119889 (119891119909 119891119910) le 120574 (1

1198874119889 (119909 119910)) (32)

for all 119909 119910 isin 119883 where 120574 [0infin) rarr [0infin) is a continuousand nondecreasing function such that 120574(119886) lt 119886 for each119886 gt 0 and 120574(0) = 0 Then 119891 has a unique fixed point

Proof If we take119892 as identitymapping on119883 thenTheorem 12follows that 119891 has a unique fixed point

Corollary 15 Let (119883 119889) be a complete metric space and let 119891and 119892 be 119877-weakly commuting self-mappings of 119883 satisfyingthe following conditions

(a) 119891(119883) sube 119892(119883)(b) 119891 or 119892 is continuous(c) 119889(119891119909 119891119910) le 120574(119889(119892119909 119892119910)) for all 119909 119910 isin 119883 where 120574

[0infin) rarr [0infin) is a continuous and nondecreasingfunction such that 120574(119886) lt 119886 for each 119886 gt 0 and 120574(0) =0


Proof If we take 119887 = 1 thenTheorem 12 follows that 119891 and 119892have a unique common fixed point

Conflict of Interests

The authors declare that they have no conflict of interests

Authorsrsquo Contribution

All authors contributed equally and significantly in writingthis paper All authors read and approved the paper

Acknowledgment

Poom Kumam and Wutiphol Sintunavarat would like tothank the Thailand Research Fund and Thammasat Uni-versity under Grant no TRG5780013 for financial supportduring the preparation of this paper

References

[1] S Czerwik ldquoNonlinear set-valued contraction mappings inb-metric spacesrdquo Atti del Seminario Matematico e FisicodellrsquoUniversita di Modena vol 46 no 2 pp 263ndash276 1998

[2] M Akkouchi ldquoCommon fixed point theorems for two selfmap-pings of a b-metric space under an implicit relationrdquo HacettepeJournal of Mathematics and Statistics vol 40 no 6 pp 805ndash8102011


[3] H AydiM-F Bota E Karapinar and SMitrovic ldquoA fixed pointtheorem for set-valued quasi-contractions in b-metric spacesrdquoFixed Point Theory and Applications vol 2012 article 88 2012

[4] M Boriceanu ldquoStrict fixed point theorems for multivaluedoperators in b-metric spacesrdquo International Journal of ModernMathematics vol 4 no 3 pp 285ndash301 2009

[5] M Boriceanu ldquoFixed point theory for multivalued generalizedcontraction on a set with two b-metricsrdquo Studia UniversitatisBabes-BolyaimdashSeries Mathematica vol 54 no 3 pp 1ndash14 2009

[6] M Boriceanu M Bota and A Petrusel ldquoMultivalued fractalsin b-metric spacesrdquo Central European Journal of Mathematicsvol 8 no 2 pp 367ndash377 2010

[7] M Bota A Molnar and C Varga ldquoOn Ekelandrsquos variationalprinciple in b-metric spacesrdquo Fixed Point Theory vol 12 no 2pp 21ndash28 2011

[8] S Czerwik ldquoContraction mappings in 119887-metric spacesrdquo ActaMathematica et Informatica Universitatis Ostraviensis vol 1 pp5ndash11 1993

[9] S Czerwik K Dlutek and S L Singh ldquoRound-off stability ofiteration procedures for operators in 119887-metric spacesrdquo Journalof Natural amp Physical Sciences vol 11 pp 87ndash94 1997

[10] N Hussain D Đoric Z Kadelburg and S Radenovic ldquoSuzuki-type fixed point results in metric type spacesrdquo Fixed PointTheory and Applications vol 2012 article 126 12 pages 2012

[11] N Hussain and M H Shah ldquoKKMmappings in cone b-metricspacesrdquoComputersampMathematics withApplications vol 62 no4 pp 1677ndash1684 2011

[12] M A Khamsi ldquoRemarks on cone metric spaces and fixed pointtheorems of contractive mappingsrdquo Fixed Point Theory andApplications Article ID 315398 7 pages 2010

[13] M A Khamsi and N Hussain ldquoKKMmappings in metric typespacesrdquoNonlinearAnalysisTheoryMethodsampApplications vol73 no 9 pp 3123ndash3129 2010

[14] P Kumam and W Sintunavarat ldquoThe existence of fixed pointtheorems for partial q-set-valued quasi-contractions in b-metric spaces and related resultsrdquo Fixed Point Theory andApplications vol 2014 article 226 2014

[15] M O Olatinwo ldquoSome results on multi-valued weakly Jungckmappings in 119887-metric spacerdquo Central European Journal ofMathematics vol 6 no 4 pp 610ndash621 2008

[16] M Pacurar ldquoSequences of almost contractions and fixed pointsin b- metric spacesrdquo Analele Universitatii de Vest TimisoaraSeria Matematica Informatica XLVIII vol 3 pp 125ndash137 2010

[17] S Phiangsungnoen W Sintunavarat and P Kumam ldquoFixedpoint results generalized Ulam-Hyers stability and well-posedness via 120572-admissiblemappings in b-metric spacesrdquo FixedPoint Theory and Applications vol 2014 article 188 2014

[18] S L Singh and B Prasad ldquoSome coincidence theorems andstability of iterative proceduresrdquoComputersampMathematics withApplications vol 55 no 11 pp 2512ndash2520 2008

[19] W Sintunavarat S Plubtieng and P Katchang ldquoFixed pointresult and applications on a b-metric space endowed with anarbitrary binary relationrdquo Fixed Point Theory and Applicationsvol 2013 article 296 2013

[20] A Aghajani M Abbas and J R Roshan ldquoCommon fixed pointof generalized weak contractive mappings in partially orderedb-metric spacesrdquo Mathematica Slovaca vol 64 no 4 pp 941ndash960 2014

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We present an example which shows that a 119887-metric on119883need not be a metric on119883 (see also [18 page 264])

Example 2 Let (119883 119889) be a metric space and 120588(119909 119910) =

(119889(119909 119910))119901 where 119901 gt 1 is a real number We show that 120588

is a 119887-metric with 119887 = 2119901minus1Obviously conditions (b1) and (b2) in Definition 1 are

satisfied Now we show that condition (b3) holds for 120588It is easy to see that if 1 lt 119901 lt infin then the convexity of

the function 119891(119909) = 119909119901 where 119909 ge 0 implies

(119886 + 119888

2)

119901

le1

2(119886119901

+ 119888119901

) (1)

and hence

(119886 + 119888)119901

le 2119901minus1

(119886119901

+ 119888119901

) (2)

Therefore for each 119909 119910 119911 isin 119883 we obtain that

120588 (119909 119910) = (119889 (119909 119910))119901

le (119889 (119909 119911) + 119889 (119911 119910))119901

le 2119901minus1

((119889 (119909 119911))119901

+ (119889 (119911 119910))119901

)

= 2119901minus1

(120588 (119909 119911) + 120588 (119911 119910))

(3)

So condition (b3) in Definition 1 holds and then 120588 is a 119887-metric coefficient 119904 = 2119901minus1

It should be noted that in Example 2 if (119883 119889) is a metricspace then (119883 120588) is not necessarily a metric space (seeExample 3)

Example 3 For example if 119883 = R and the usual Euclideanmetric 119889 119883 times 119883 rarr R

+is defined by

119889 (119909 119910) =1003816100381610038161003816119909 minus 119910

1003816100381610038161003816 (4)

for all 119909 119910 isin 119883 then 120588(119909 119910) = |119909minus119910|2 is a 119887-metric onRwith119887 = 2 but is not a metric onR because the triangle inequalitydoes not hold

Before stating and proving our results we present somedefinition and proposition in 119887-metric space We recall firstthe notions of convergence closedness and completeness ina 119887-metric space

Definition 4 (see [6]) Let (119883 119889) be a 119887-metric space Then asequence 119909

119899 in119883 is called

(i) convergent if and only if there exists 119909 isin 119883 such that119889(119909119899 119909) rarr 0 as 119899 rarr +infin and in this case we write

lim119899rarrinfin

119909119899= 119909

(ii) Cauchy if and only if 119889(119909119899 119909119898) rarr 0 as 119899119898 rarr +infin

Proposition 5 (Remark 21 in [6]) In a 119887-metric space (119883 119889)the following assertions hold

(i) a convergent sequence has a unique limit

(ii) each convergent sequence is Cauchy(iii) in general a 119887-metric is not continuous

Definition 6 (see [6]) The 119887-metric space (119883 119889) is completeif every Cauchy sequence in119883 converges

It should be noted that in general a 119887-metric function119889 for119887 gt 1 is not jointly continuous in all the two of its variables

Since in general a 119887-metric is not continuous we need thefollowing simple lemma about the 119887-convergent sequences

Lemma 7 (see [20]) Let (119883 119889) be a 119887-metric space withcoefficient 119887 ge 1 and suppose that 119909

119899 and 119910

119899 are 119887-

convergent to 119909 119910 respectively then one has

1

1198872119889 (119909 119910) le lim inf

119899rarrinfin

119889 (119909119899 119910119899) le lim sup119899rarrinfin

119889 (119909119899 119910119899)

le 1198872

119889 (119909 119910)

(5)

In particular if 119909 = 119910 then one has lim119899rarrinfin

119889(119909119899 119910119899) = 0

Moreover for each 119911 isin 119883 one has

1

119887119889 (119909 119911) le lim inf

119899rarrinfin

119889 (119909119899 119911) le lim sup

119899rarrinfin

119889 (119909119899 119911) le 119887119889 (119909 119911)

(6)

3 Common Fixed Point Results

Definition 8 Let 119891 and 119892 be mappings from a 119887-metric space(119883 119889) into itself The mappings 119891 and 119892 are said to be weaklycommuting if

119889 (119891119892119909 119892119891119909) le 119889 (119891119909 119892119909) (7)

for each 119909 in119883

Definition 9 Let 119891 and 119892 be mappings from a 119887-metric space(119883 119889) into itself The mappings 119891 and 119892 are said to be 119877-weakly commuting if there exists some positive real number119877 such that

119889 (119891119892119909 119892119891119909) le 119877119889 (119891119909 119892119909) (8)

for each 119909 in119883

Remark 10 Weak commutativity implies 119877-weak commu-tativity in 119887-metric space However 119877-weak commutativityimplies weak commutativity only when 119877 le 1

Example 11 Let 119883 = R and 119889 119883 times 119883 rarr R+defined as

follows

119889 (119909 119910) = (119909 minus 119910)2 (9)

for all 119909 119910 isin 119883 Then (119883 119889) is a 119887-metric space Define 119891119909 =2119909 minus 1 and 119892119909 = 1199092 Then

119889 (119891119892119909 119892119891119909) = 4 (119909 minus 1)4

= 4119889 (119891119909 119892119909) gt 119889 (119891119909 119892119909) (10)

Therefore for 119877 = 4 119891 and 119892 are 119877-weakly commuting But119891 and 119892 are not weakly commuting




4)119889(119892119909 119892119910)) for all 119909 119910 isin 119883




1199091in119883 such that119891119909


119899+1such that

119891119909119899= 119892119909119899+1


119889 (119891119909119899 119891119909119899+1

) le 120574 (1

1198874119889 (119892119909119899 119892119909119899+1

))

= 120574 (1

1198874119889 (119891119909119899minus1

119891119909119899))

le1

1198874119889 (119891119909119899minus1

119891119909119899)

le 119889 (119891119909119899minus1

119891119909119899)

(11)

This implies that

119888119899 = 119889 (119891119909

119899 119891119909119899+1

) (12)


119886 le 120574 (1

1198874119886) lt

1

1198874119886 le 119886 (13)


lim119899rarrinfin

119888119899= lim119899rarrinfin

119889 (119891119909119899 119891119909119899+1

) = 0 (14)





119889119896= 119889 (119910

119899(119896) 119910119898(119896)

) ge 120576 for 119896 = 1 2 (15)

We may assume that

119889 (119910119899(119896)

119910119898(119896)minus1

) lt 120576 (16)


120576 le 119889119896le 119887 [119889 (119910

119899(119896) 119910119898(119896)minus1

) + 119889 (119910119898(119896)minus1

119910119898(119896)

)]

lt 119887120576 + 119887119888119898(119896)minus1

le 119887120576 + 119887119888119896

(17)


119889119896le 119887120576 Also notice

119889119896= 119889 (119910

119899(119896) 119910119898(119896)

)

le 119887119889 (119910119899(119896)

119910119899(119896)+1

)

+ 1198872

[119889 (119910119899(119896)+1

119910119898(119896)+1

) + 119889 (119910119898(119896)+1

119910119898(119896)

)]

le 119887119888119896+ 1198872

120574 (1

1198874119889 (119910119899(119896)

119910119898(119896)

)) + 1198872

119888119896

= 119887119888119896+ 1198872

120574 (1

1198874119889119896) + 1198872

119888119896

(18)



119889119896le 119887 lim119896rarrinfin

119888119896+ 1198872

120574 (1


119889119896) + 1198872 lim119896rarrinfin

119888119896

(19)

and thus

120576 le 1198872

120574 (1

1198873120576) lt

1

119887120576 (20)





119891119891119909119899= 119891119911 and

lim119899rarrinfin



119889 (119891119892119909119899 119892119891119909119899) le 119877119889 (119891119909

119899 119892119909119899) (21)


1

1198872119889(119891119911 lim sup

119899rarrinfin


119889 (119891119892119909119899 119892119891119909119899)


119889 (119891119909119899 119892119909119899)

le 1198771198872

119889 (119911 119911) = 0

(22)

Similarly

1

1198872119889 (119891119911 lim inf

119899rarrinfin

119892119891119909119899) = 0 (23)


119892119891119909119899= 119891119911 We now prove that 119911 =


1

119887119889 (119891119911 119891119909


119889 (119891119891119909119899 119891119909119898)


1

1198874119889 (119892119891119909

119899 119892119909119898))

le 120574 (1

1198873119889 (119891119911 119892119909

119898))

(24)



1

1198872119889 (119891119911 119911) le lim sup

119898rarrinfin

1

119887119889 (119891119911 119891119909

119898)


1

1198873119889 (119891119911 119892119909

119898))

le 120574 (1

1198872119889 (119891119911 119911)) lt

1

1198872119889 (119891119911 119911)

(25)


1in 119883 such that 119911 = 119891119911 = 119892119911

1 Now we

have

119889 (119891119891119909119899 1198911199111) le 120574 (

1

1198874119889 (119892119891119909

119899 1198921199111)) (26)


1

119887119889 (119891119911 119891119911


119889 (119891119891119909119899 119891119911)


1

1198874119889 (119892119891119909

119899 1198921199111))

le 120574 (1

1198873119889 (119891119911 119892119911

1))

= 0

(27)


1198911199111= 1198921199111 Also

119889 (119891119911 119892119911) = 119889 (1198911198921199111 1198921198911199111) le 119877119889 (119891119911

1 1198921199111) = 0 (28)



119889 (119911 1199111015840

) = 119889 (119891119911 1198911199111015840

) le 120574 (1

1198874119889 (119892119911 119892119911

1015840

))

= 120574 (1

1198874119889 (119911 119911

1015840

)) lt1

1198874119889 (119911 119911

1015840

) le 119889 (119911 1199111015840

)

(29)





119889 (119909 119910) = (119909 minus 119910)2 (30)


119889 (119891119909 119891119910) le 120574 (1

16119889 (119892119909 119892119910)) (31)



119889 (119891119909 119891119910) le 120574 (1

1198874119889 (119909 119910)) (32)












Acknowledgment


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












4)119889(119892119909 119892119910)) for all 119909 119910 isin 119883




1199091in119883 such that119891119909


119899+1such that

119891119909119899= 119892119909119899+1


119889 (119891119909119899 119891119909119899+1

) le 120574 (1

1198874119889 (119892119909119899 119892119909119899+1

))

= 120574 (1

1198874119889 (119891119909119899minus1

119891119909119899))

le1

1198874119889 (119891119909119899minus1

119891119909119899)

le 119889 (119891119909119899minus1

119891119909119899)

(11)

This implies that

119888119899 = 119889 (119891119909

119899 119891119909119899+1

) (12)


119886 le 120574 (1

1198874119886) lt

1

1198874119886 le 119886 (13)


lim119899rarrinfin

119888119899= lim119899rarrinfin

119889 (119891119909119899 119891119909119899+1

) = 0 (14)





119889119896= 119889 (119910

119899(119896) 119910119898(119896)

) ge 120576 for 119896 = 1 2 (15)

We may assume that

119889 (119910119899(119896)

119910119898(119896)minus1

) lt 120576 (16)


120576 le 119889119896le 119887 [119889 (119910

119899(119896) 119910119898(119896)minus1

) + 119889 (119910119898(119896)minus1

119910119898(119896)

)]

lt 119887120576 + 119887119888119898(119896)minus1

le 119887120576 + 119887119888119896

(17)


119889119896le 119887120576 Also notice

119889119896= 119889 (119910

119899(119896) 119910119898(119896)

)

le 119887119889 (119910119899(119896)

119910119899(119896)+1

)

+ 1198872

[119889 (119910119899(119896)+1

119910119898(119896)+1

) + 119889 (119910119898(119896)+1

119910119898(119896)

)]

le 119887119888119896+ 1198872

120574 (1

1198874119889 (119910119899(119896)

119910119898(119896)

)) + 1198872

119888119896

= 119887119888119896+ 1198872

120574 (1

1198874119889119896) + 1198872

119888119896

(18)



119889119896le 119887 lim119896rarrinfin

119888119896+ 1198872

120574 (1


119889119896) + 1198872 lim119896rarrinfin

119888119896

(19)

and thus

120576 le 1198872

120574 (1

1198873120576) lt

1

119887120576 (20)





119891119891119909119899= 119891119911 and

lim119899rarrinfin



119889 (119891119892119909119899 119892119891119909119899) le 119877119889 (119891119909

119899 119892119909119899) (21)


1

1198872119889(119891119911 lim sup

119899rarrinfin


119889 (119891119892119909119899 119892119891119909119899)


119889 (119891119909119899 119892119909119899)

le 1198771198872

119889 (119911 119911) = 0

(22)

Similarly

1

1198872119889 (119891119911 lim inf

119899rarrinfin

119892119891119909119899) = 0 (23)


119892119891119909119899= 119891119911 We now prove that 119911 =


1

119887119889 (119891119911 119891119909


119889 (119891119891119909119899 119891119909119898)


1

1198874119889 (119892119891119909

119899 119892119909119898))

le 120574 (1

1198873119889 (119891119911 119892119909

119898))

(24)



1

1198872119889 (119891119911 119911) le lim sup

119898rarrinfin

1

119887119889 (119891119911 119891119909

119898)


1

1198873119889 (119891119911 119892119909

119898))

le 120574 (1

1198872119889 (119891119911 119911)) lt

1

1198872119889 (119891119911 119911)

(25)


1in 119883 such that 119911 = 119891119911 = 119892119911

1 Now we

have

119889 (119891119891119909119899 1198911199111) le 120574 (

1

1198874119889 (119892119891119909

119899 1198921199111)) (26)


1

119887119889 (119891119911 119891119911


119889 (119891119891119909119899 119891119911)


1

1198874119889 (119892119891119909

119899 1198921199111))

le 120574 (1

1198873119889 (119891119911 119892119911

1))

= 0

(27)


1198911199111= 1198921199111 Also

119889 (119891119911 119892119911) = 119889 (1198911198921199111 1198921198911199111) le 119877119889 (119891119911

1 1198921199111) = 0 (28)



119889 (119911 1199111015840

) = 119889 (119891119911 1198911199111015840

) le 120574 (1

1198874119889 (119892119911 119892119911

1015840

))

= 120574 (1

1198874119889 (119911 119911

1015840

)) lt1

1198874119889 (119911 119911

1015840

) le 119889 (119911 1199111015840

)

(29)





119889 (119909 119910) = (119909 minus 119910)2 (30)


119889 (119891119909 119891119910) le 120574 (1

16119889 (119892119909 119892119910)) (31)



119889 (119891119909 119891119910) le 120574 (1

1198874119889 (119909 119910)) (32)












Acknowledgment


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1

1198872119889 (119891119911 119911) le lim sup

119898rarrinfin

1

119887119889 (119891119911 119891119909

119898)


1

1198873119889 (119891119911 119892119909

119898))

le 120574 (1

1198872119889 (119891119911 119911)) lt

1

1198872119889 (119891119911 119911)

(25)


1in 119883 such that 119911 = 119891119911 = 119892119911

1 Now we

have

119889 (119891119891119909119899 1198911199111) le 120574 (

1

1198874119889 (119892119891119909

119899 1198921199111)) (26)


1

119887119889 (119891119911 119891119911


119889 (119891119891119909119899 119891119911)


1

1198874119889 (119892119891119909

119899 1198921199111))

le 120574 (1

1198873119889 (119891119911 119892119911

1))

= 0

(27)


1198911199111= 1198921199111 Also

119889 (119891119911 119892119911) = 119889 (1198911198921199111 1198921198911199111) le 119877119889 (119891119911

1 1198921199111) = 0 (28)



119889 (119911 1199111015840

) = 119889 (119891119911 1198911199111015840

) le 120574 (1

1198874119889 (119892119911 119892119911

1015840

))

= 120574 (1

1198874119889 (119911 119911

1015840

)) lt1

1198874119889 (119911 119911

1015840

) le 119889 (119911 1199111015840

)

(29)





119889 (119909 119910) = (119909 minus 119910)2 (30)


119889 (119891119909 119891119910) le 120574 (1

16119889 (119892119909 119892119910)) (31)



119889 (119891119909 119891119910) le 120574 (1

1198874119889 (119909 119910)) (32)












Acknowledgment


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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