research article common fixed point theorems for...

Hindawi Publishing CorporationJournal of OperatorsVolume 2013 Article ID 391474 5 pageshttpdxdoiorg1011552013391474

Research ArticleCommon Fixed Point Theorems for Conversely CommutingMappings Using Implicit Relations

Sunny Chauhan1 and Huma Sahper2

1 Near Nehru Training Centre H No 274 Nai Basti B-14 Bijnor Uttar Pradesh 246701 India2Department of Applied Mathematics Z H College of Engineering and Technology Aligarh Muslim University AligarhUttar Pradesh 202002 India

Correspondence should be addressed to Sunny Chauhan sungkvgmailcom

Received 24 August 2013 Accepted 25 October 2013

Academic Editor Haiyan Wang

Copyright copy 2013 S Chauhan and H Sahper 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

The object of this paper is to utilize the notion of conversely commuting mappings due to Lu (2002) and prove some common fixedpoint theorems in Menger spaces via implicit relations We give some examples which demonstrate the validity of the hypothesesand degree of generality of our main results

1 Introduction

In 1986 Jungck [1] introduced the notion of compatiblemappings in metric space Most of the common fixed pointtheorems for contraction mappings invariably require acompatibility condition besides continuity of at least one ofthe mappings Later on Jungck and Rhoades [2] studied thenotion of weakly compatible mappings and utilized it as atool to improve commutativity conditions in common fixedpoint theorems Many mathematicians proved several fixedpoint results in Menger spaces (see eg [3ndash9]) In 2002Lu [10] presented the concept of the converse commutingmappings as a reverse process of weakly compatiblemappingsand proved common fixed point theorems for single-valuedmappings in metric spaces (also see [11]) Recently Pathakand Verma [12 13] Chugh et al [14] and Chauhan et al [15]proved some interesting common fixed point theorems forconverse commuting mappings

In this paper we prove some unique common fixed pointtheorems for two pairs of converse commuting mappings inMenger spaces by using implicit relations

2 Preliminaries

Definition 1 (see [16]) A 119905-norm is a function Δ [0 1] times

[0 1] rarr [0 1] satisfying

(T1) Δ(119886 1) = 119886 Δ(0 0) = 0(T2) Δ(119886 119887) = Δ(119887 119886)(T3) Δ(119888 119889) ge Δ(119886 119887) for 119888 ge 119886 119889 ge 119887(T4) Δ(Δ(119886 119887) 119888) = Δ(119886 Δ(119887 119888)) for all 119886 119887 119888 in [0 1]

Examples of 119905-norms are Δ(119886 119887) = min119886 119887 Δ(119886 119887) =119886119887 and Δ(119886 119887) = max119886 + 119887 minus 1 0

Definition 2 (see [16]) A real valued function 119891 on theset of real numbers is called a distribution function if it isnondecreasing left continuous with inf

119906isinR119891(119906) = 0 andsup119906isinR119891(119906) = 1We shall denote by I the set of all distribution functions

defined on (minusinfininfin) while 119867(119905) will always denote thespecific distribution function defined by

119867(119905) = 0 if 119905 le 01 if 119905 gt 0

(1)

If 119883 is a nonempty set F 119883 times 119883 rarr I is called aprobabilistic distance on 119883 and the value of F at (119909 119910) isin119883 times 119883 is represented by 119865

119909119910

Definition 3 (see [17]) A probabilistic metric space is anordered pair (119883F) where 119883 is a nonempty set of elements

2 Journal of Operators

and F is a probabilistic distance satisfying the followingconditions for all 119909 119910 119911 isin 119883 and 119905 119904 gt 0

(1) 119865119909119910(119905) = 1 for all 119905 gt 0 if and only if 119909 = 119910

(2) 119865119909y(119905) = 119865119910119909(119905)

(3) 119865119909119910(0) = 0

(4) if 119865119909119910(119905) = 1 and 119865

119910119911(119904) = 1 then 119865

119909119911(119905 + 119904) = 1 for

all 119909 119910 119911 isin 119883 and 119905 119904 ge 0

Every metric space (119883 119889) can always be realized as aprobabilistic metric space by considering F 119883 times 119883 rarr Idefined by 119865

119909119910(119905) = 119867(119905 minus 119889(119909 119910)) for all 119909 119910 isin 119883 and

119905 isin 119877 So probabilistic metric spaces offer a wider frameworkthan that of metric spaces and are better suited to cover evenwider statistical situations that is every metric space can beregarded as a probabilistic metric space of a special kind

Definition 4 (see [16]) A Menger space (119883F Δ) is a tripletwhere (119883F) is a probabilisticmetric space andΔ is a 119905-normsatisfying the following condition

119865119909119910(119905 + 119904) ge Δ (119865

119909119911(119905) 119865119911119910(119904)) (2)

for all 119909 119910 119911 isin 119883 and 119905 119904 ge 0

Definition 5 (see [2]) A pair (119860 119878) of self-mappings definedon a nonempty set 119883 is said to be weakly compatible(or coincidentally commuting) if they commute at theircoincidence points that is if 119860119909 = 119878119909 for some 119909 isin 119883 then119860119878119909 = 119878119860119909

Definition 6 (see [10]) A pair (119860 119878) of self-mappings definedon a nonempty set 119883 is called conversely commuting if forall 119909 isin 119883 119860119878119909 = 119878119860119909 implies 119860119909 = 119878119909

Definition 7 (see [10]) Let 119860 and 119878 be self-mappings of anonempty set 119883 A point 119909 isin 119883 is called commuting pointof 119860 and 119878 if 119860119878119909 = 119878119860119909

Lemma 8 (see [18]) Let (119883F Δ) be a Menger space If thereexists a constant 119896 isin (0 1) such that

119865119909119910(119896119905) ge 119865

119909y (119905) (3)

for all 119905 gt 0 with fixed 119909 119910 isin 119883 then 119909 = 119910

3 Implicit Relations

In 2005 Singh and Jain [19] studied an implicit functionand obtained some fixed point results in framework of fuzzymetric spaces

Let Φ be the set of all real continuous functions 120601

[0 1]4

rarr R nondecreasing in first argument and satisfyingthe following conditions

(120601-1) For 119906 V ge 0 120601(119906 V 119906 V) ge 0 or 120601(119906 V V 119906) ge 0

implies that 119906 ge V(120601-2) 120601(119906 119906 1 1) ge 0 implies that 119906 ge 1

Example 9 Define 120601 [0 1]4 rarr R as 120601(1199051 1199052 1199053 1199054) = 18119905

1minus

161199052+ 81199053minus 101199054 Then 120601 isin Φ

Since then Imdad and Ali [20] used the following class ofimplicit functions for the existence of a common fixed pointdue to Popa [21] Many authors proved a number of commonfixed point theorems using the notion of implicit relation ondifferent spaces (see eg [22ndash29])

Let Ψ denote the family of all continuous functions 120595

[0 1]4

rarr R satisfying the following conditions

(120595-1) For every 119906 gt 0 V ge 0 with 120595(119906 V 119906 V) ge 0 or120595(119906 V V 119906) ge 0 we have 119906 gt V

(120595-2) 120595(119906 119906 1 1) lt 0 for all 119906 gt 0

Example 10 Define 120595 [0 1]4 rarr R as 120595(1199051 1199052 1199053 1199054) = 1199051minus

120593(min1199052 1199053 1199054) where 120593 [0 1] rarr [0 1] is a continuous

function such that 120593(119904) gt 119904 for 0 lt 119904 lt 1


119896min1199052 1199053 1199054 where 119896 gt 1


1198961199052minusmin119905

3 1199054 where 119896 gt 0


1198861199052minus 1198871199053minus 1198881199054 where 119886 gt 1 and 119887 119888 ge 0 (119887 119888 = 1)


1198861199052minus 119887(1199053+ 1199054) where 119886 gt 1 and 0 le 119887 lt 1


1minus

119896119905211990531199054 where 119896 gt 1

In 2011 Gopal et al [30] showed that the above-mentioned classes of functions Φ and Ψ are independentclasses

4 Main Results

First we prove a unique common fixed point theorem for twopairs of self-mappings satisfying a class of implicit functionΨ

Theorem 16 Let 119860 119861 119878 and 119879 be four self-mappings ofa Menger space (119883F Δ) where Δ is a continuous 119905-normand the pairs (119860 119878) and (119861 119879) are conversely commutingrespectively and satisfy the following conditions

120595 (119865119860119909119861119910

(119905) 119865119878119909119879119910

(119905) 119865119860119909119878119909

(119905) 119865119861119910119879119910

(119905)) ge 0 (4)

for all 119909 119910 isin 119883 119905 gt 0 and120595 isin Ψ If119860 and 119878 have a commutingpoint and 119861 and 119879 have a commuting point then 119860 119861 119878 and119879 have a unique common fixed point in 119883

Proof Let 119906 be the commuting point of119860 and 119878 Then119860119878119906 =119878119860119906 And let V be the commuting point of 119861 and 119879 Then119861119879V = 119879119861V Since 119860 and 119878 are conversely commuting wehave 119860119906 = 119878119906 Since 119861 and 119879 are conversely commutingwe have 119861V = 119879V Hence 119860119860119906 = 119860119878119906 = 119878119860119906 = 119878119878119906 and119861119861V = 119861119879V = 119879119861V = 119879119879V

Journal of Operators 3

(i) We claim that 119860119906 = 119861V On using (4) with 119909 = 119906119910 = V we get

120595 (119865119860119906119861V (119905) 119865119878119906119879V (119905) 119865119860119906119878119906 (119905) 119865119861V119879V (119905)) ge 0 (5)

or equivalently

120595 (119865119860119906119861V (119905) 119865119860119906119861V (119905) 1 1) ge 0 (6)

Hence for 119865119860119906119861V(119905) = 1 for all 119905 gt 0 we have 119860119906 =

119861V Thus 119860119906 = 119878119906 = 119861V = 119879V(ii) Now we show that 119860119906 is a fixed point of mapping 119860

In order to establish this using (4)with119909 = 119860119906119910 = Vwe have

120595 (119865119860119860119906119861V (119905) 119865119878119860119906119879V (119905) 119865119860119860119906119878119860119906 (119905) 119865119861V119879V (119905)) ge 0 (7)

and so

120595 (119865119860119860119906119860119906

(119905) 119865119860119860119906119860119906

(119905) 1 1) ge 0 (8)

Hence for 119865119860119860119906119860119906

(119905) = 1 for all 119905 gt 0 we get119860119860119906 = 119860119906 Similarly we show that 119861V = 119861119861V Onusing (4) with 119909 = 119906 119910 = 119861V we have

120595 (119865119860119906119861119861V (119905) 119865119878119906119879119861V (119905) 119865119860119906119878119906 (119905) 119865119861119861V119879119861V (119905)) ge 0 (9)

or equivalently

120595 (119865119861V119861119861V (119905) 119865119861V119861119861V (119905) 1 1) ge 0 (10)

Thus 119865119860119860119906119860119906

(119905) = 1 for all 119905 gt 0 and we obtain119861119861V = 119861V Since 119860119906 = 119861V we have 119860119906 = 119861V = 119861119861V =119861119860119906 which shows that 119860119906 is a fixed point of themapping 119861 On the other hand 119860119906 = 119861V = 119861119861V =119879119861V = 119879119860119906 and 119860119906 = 119860119860119906 = 119860119878119906 = 119878119860119906 Hence119860119906(= 119908 isin 119883) is a common fixed point of 119860 119861 119878 and119879

(iii) For the uniqueness of common fixed point we use (4)with 119909 = 119908 and 119910 = where is another commonfixed point of the mappings 119860 119861 119878 and 119879 Now wehave

120595 (119865119860119908119861

(119905) 119865119878119908119879

(119905) 119865119860119908119878119908

(119905) 119865119861119879

(119905)) ge 0 (11)

and so

120595 (119865119908(119905) 119865119908(119905) 1 1) ge 0 (12)

Hence we get 119908 = Therefore 119908 is a uniquecommon fixed point of the mappings 119860 119861 119878 and 119879

Now we give an example which illustrates Theorem 16

Example 17 Let 119883 = [1infin) with the metric 119889 defined by119889(119909 119910) = |119909 minus 119910| and for each 119905 isin [0 1] define

119865119909119910(119905) =

119905

119905 +1003816100381610038161003816119909 minus 119910

1003816100381610038161003816

if 119905 gt 0

0 if 119905 = 0(13)

for all 119909 119910 isin 119883 Define Δ(119886 119887) = min119886 119887 Clearly (119883F Δ)is a Menger space Let 120595 [0 1]

4

rarr R as 120595(1199051 1199052 1199053 1199054) =

1199051minus120593(min119905

2 1199053 1199054) with 120593(119904) = radic119904 for 0 lt 119904 lt 1 Define the

self-mappings 119860 119861 119878 and 119879 by

119860 (119909) = 2119909 minus 1 if119909 lt 21 if 119909 ge 2

119878 (119909) = 1199092

if 119909 lt 2119909 + 3 if 119909 ge 2

119861 (119909) = 2119909 minus 1 if119909 lt 22 if 119909 ge 2

119879 (119909) = 31199092

minus 2 if 119909 lt 21199092

+ 1 if 119909 ge 2

(14)

Hence the pairs (119860 119878) and (119861 119879) are conversely commut-ing and 1 is a unique common fixed point of the mappings119860 119861 119878 and 119879

Corollary 18 The conclusions of Theorem 16 remain true ifcondition (4) is replaced by one of the following conditions forall 119909 119910 isin 119883

119865119860119909119861119910

(119905) ge 120593 (min 119865119878119909119879119910

(119905) 119865119860119909119878119909

(119905) 119865119861119910119879119910

(119905))

(15)

where 120593 [0 1] rarr [0 1] is a continuous function such that120593(119904) gt 119904 for all 0 lt 119904 lt 1

119865119860119909119861119910

(119905) ge 119896 (min 119865119878119909119879119910

(119905) 119865119860119909119878119909

(119905) 119865119861119910119879119910

(119905))

(16)

where 119896 gt 1

119865119860119909119861119910

(119905) ge 119896119865119878119909119879119910

(119905) +min 119865119860119909119878119909

(119905) 119865119861119910119879119910

(119905) (17)

where 119896 gt 0

119865119860119909119861119910

(119905) ge 119886119865119878119909119879119910

(119905) + 119887119865119860119909119878119909

(119905) + 119888119865119861119910119879119910

(119905) (18)

where 119886 gt 1 and 119887 119888 ge 0 (119887 119888 = 1)

119865119860119909119861119910

(119905) ge 119886119865119878119909119879119910

(119905) + 119887 [119865119860119909119878119909

(119905) + 119865119861119910119879119910

(119905)] (19)

where 119886 gt 1 and 0 le 119887 lt 1

119865119860119909119861119910

(119905) ge 119896119865119878119909119879119910

(119905) 119865119860119909119878119909

(119905) 119865119861119910119879119910

(119905) (20)

where 119896 gt 1

Proof The proof of each inequality (15)ndash(20) easily followsfromTheorem 16 in view of Examples 10ndash15

Now we state a unique common fixed point theoremsatisfying a class of implicit function Φ


Theorem 19 Let 119860 119861 119878 and 119879 be four self-mappings on aMenger space (119883F Δ) where Δ is a continuous t-norm thepairs (119860 119878) and (119861 119879) are conversely commuting respectivelyand satisfying

120601 (119865119860119909119861119910

(119896119905) 119865119878119909119879119910

(119905) 119865119860119909119878119909

(119905) 119865119861119910119879119910

(119896119905)) ge 0

120601 (119865119860119909119861119910

(119896119905) 119865119878119909119879119910

(119905) 119865119860119909119878119909

(119896119905) 119865119861119910119879119910

(119905)) ge 0

(21)

for all 119909 119910 isin 119883 119905 gt 0 119896 isin (0 1) and 120601 isin Φ If 119860 and 119878 havea commuting point and 119861 and 119879 have a commuting point then119860 119861 119878 and 119879 have a unique common fixed point in 119883

Proof The proof of this theorem can be completed on thelines of the proof ofTheorem 16 (in view of Lemma 8) hencedetails are omitted

Example 20 In the setting of Example 17 define120601 [0 1]4 rarrR as 120601(119905

1 1199052 1199053 1199054) = 18119905

1minus 16119905

2minus 81199053+ 10119905

4besides

retaining the restTherefore all the conditions ofTheorem 19are satisfied for some fixed 119896 isin (0 1) and 1 is a uniquecommon fixed point of the mappings 119860 119861 119878 and 119879

By choosing 119860 119861 119878 and 119879 suitably we can deducecorollaries involving two as well as three self-mappings Forthe sake of naturality we only derive the following corollary(due toTheorem 16) involving a pair of self-mappings

Corollary 21 Let 119860 and 119878 be two self-mappings of a Mengerspace (119883F Δ) where Δ is a continuous t-norm and themappings 119860 and 119878 are conversely commuting satisfying

120595 (119865119860119909119860119910

(119905) 119865119878119909119878119910

(119905) 119865119860119909119878119909

(119905) 119865119860119910119878119910

(119905)) ge 0 (22)

for all 119909 119910 isin 119883 119905 gt 0 and120595 isin Ψ If119860 and 119878 have a commutingpoint then 119860 and 119878 have a unique common fixed point in 119883

Conflict of Interests

The authors declare that they have no conflict of interests

References

[1] G Jungck ldquoCompatible mappings and common fixed pointsrdquoInternational Journal of Mathematics and Mathematical Sci-ences vol 9 no 4 pp 771ndash779 1986

[2] G Jungck and B E Rhoades ldquoFixed points for set valuedfunctions without continuityrdquo Indian Journal of Pure andApplied Mathematics vol 29 no 3 pp 227ndash238 1998

[3] S Chauhan and B D Pant ldquoCommon fixed point theoremfor weakly compatible mappings in Menger spacerdquo Journal ofAdvanced Research in Pure Mathematics vol 3 no 2 pp 107ndash119 2011

[4] M Imdad J Ali and M Tanveer ldquoCoincidence and commonfixed point theorems for nonlinear contractions in Menger PMspacesrdquo Chaos Solitons amp Fractals vol 42 no 5 pp 3121ndash31292009

[5] D Mihet ldquoA note on a common fixed point theorem inprobabilistic metric spacesrdquo Acta Mathematica Hungarica vol125 no 1-2 pp 127ndash130 2009

[6] B D Pant and S Chauhan ldquoA contraction theorem in Mengerspacerdquo Tamkang Journal of Mathematics vol 42 no 1 pp 59ndash68 2011

[7] B D Pant and S Chauhan ldquoCommon fixed point theorems fortwo pairs of weakly compatible mappings inMenger spaces andfuzzy metric spacesrdquo Scientific Studies and Research vol 21 no2 pp 81ndash96 2011

[8] B D Pant S Chauhan and Q Alam ldquoCommon fixed pointtheorem in probabilistic metric spacerdquo Kragujevac Journal ofMathematics vol 35 no 3 pp 463ndash470 2011

[9] R Saadati D OrsquoRegan S M Vaezpour and J K Kim ldquoGen-eralized distance and common fixed point theorems in Mengerprobabilistic metric spacesrdquo Iranian Mathematical Society vol35 no 2 pp 97ndash117 2009

[10] Z X Lu ldquoCommon fixed points for converse commuting self-maps on a metric spacerdquo Acta Analysis Functionalis Applicatavol 4 no 3 pp 226ndash228 2002 (Chinese)

[11] Q K Liu and X Q Hu ldquoSome new common fixed pointtheorems for converse commuting multi-valued mappings insymmetric spaces with applicationsrdquoNonlinear Analysis Forumvol 10 no 1 pp 97ndash104 2005

[12] H K Pathak and R K Verma ldquoIntegral type contractivecondition for converse commuting mappingsrdquo InternationalJournal ofMathematical Analysis vol 3 no 21ndash24 pp 1183ndash11902009

[13] HK Pathak andRKVerma ldquoAn integral type implicit relationfor converse commuting mappingsrdquo International Journal ofMathematical Analysis vol 3 no 21ndash24 pp 1191ndash1198 2009

[14] R Chugh Sumitra and M Alamgir Khan ldquoCommon fixedpoint theorems for converse commuting maps in fuzzy metricspacesrdquo Journal for Theory and Applications vol 6 no 37ndash40pp 1845ndash1851 2011

[15] S Chauhan M A Khan and W Sintunavarat ldquoFixed pointsof converse commuting mappings using an implicit relationrdquoHonam Mathematical Journal vol 35 no 2 pp 109ndash117 2013

[16] B Schweizer and A Sklar ldquoStatistical metric spacesrdquo PacificJournal of Mathematics vol 10 pp 313ndash334 1960

[17] K Menger ldquoStatistical metricsrdquo Proceedings of the NationalAcademy of Sciences of the United States of America vol 28 pp535ndash537 1942

[18] S NMishra ldquoCommon fixed points of compatible mappings inPM-spacesrdquo Mathematica Japonica vol 36 no 2 pp 283ndash2891991

[19] B Singh and S Jain ldquoSemicompatibility and fixed point theo-rems in fuzzymetric space using implicit relationrdquo InternationalJournal of Mathematics and Mathematical Sciences no 16 pp2617ndash2629 2005

[20] M Imdad and J Ali ldquoA general fixed point theorems infuzzy metric spaces via implicit functionrdquo Journal of AppliedMathematics amp Informatics vol 26 no 3-4 pp 591ndash603 2008

[21] V Popa ldquoA fixed point theorem for mapping in d-completetopological spacesrdquo Mathematica Moravica vol 3 pp 43ndash481999

[22] S Chauhan M Imdad and C Vetro ldquoUnified metrical com-mon fixed point theorems in 2-metric spaces via an implicitrelationrdquo Journal of Operators vol 2013 Article ID 186910 11pages 2013

[23] S Chauhan M A Khan and S Kumar ldquoUnified fixed pointtheorems in fuzzy metric spaces via common limit rangepropertyrdquo Journal of Inequalities and Applications vol 2013article 182 17 pages 2013


[24] S Chauhan and B D Pant ldquoFixed points of weakly compatiblemappings using common (EA) like propertyrdquo Le Matematichevol 68 no 1 pp 99ndash116 2013

[25] M Imdad and S Chauhan ldquoEmploying common limit rangeproperty to prove unified metrical common fixed point the-oremsrdquo International Journal of Analysis vol 2013 Article ID763261 10 pages 2013

[26] S Kumar and S Chauhan ldquoCommon fixed point theoremsusing implicit relation and property (EA) in fuzzy metricspacesrdquoAnnals of FuzzyMathematics and Informatics vol 5 no1 pp 107ndash114 2013

[27] S Kumar and B D Pant ldquoCommon fixed point theorems inprobabilistic metric spaces using implicit relation and property(EA)rdquo Bulletin of the Allahabad Mathematical Society vol 25no 2 pp 223ndash235 2010

[28] V Popa andD Turkoglu ldquoSome fixed point theorems for hybridcontractions satisfying an implicit relationrdquo Studii si CercetariStiintifice no 8 pp 75ndash86 1998

[29] S Sharma and B Deshpande ldquoOn compatiblemappings satisfy-ing an implicit relation in common fixed point considerationrdquoTamkang Journal of Mathematics vol 33 no 3 pp 245ndash2522002

[30] DGopalM Imdad andCVetro ldquoImpact of commonproperty(EA) on fixed point theorems in fuzzy metric spacesrdquo FixedPoint Theory and Applications vol 2011 Article ID 297360 14pages 2011

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Stochastic AnalysisInternational Journal of


and F is a probabilistic distance satisfying the followingconditions for all 119909 119910 119911 isin 119883 and 119905 119904 gt 0

(1) 119865119909119910(119905) = 1 for all 119905 gt 0 if and only if 119909 = 119910

(2) 119865119909y(119905) = 119865119910119909(119905)

(3) 119865119909119910(0) = 0

(4) if 119865119909119910(119905) = 1 and 119865

119910119911(119904) = 1 then 119865

119909119911(119905 + 119904) = 1 for

all 119909 119910 119911 isin 119883 and 119905 119904 ge 0

Every metric space (119883 119889) can always be realized as aprobabilistic metric space by considering F 119883 times 119883 rarr Idefined by 119865

119909119910(119905) = 119867(119905 minus 119889(119909 119910)) for all 119909 119910 isin 119883 and

119905 isin 119877 So probabilistic metric spaces offer a wider frameworkthan that of metric spaces and are better suited to cover evenwider statistical situations that is every metric space can beregarded as a probabilistic metric space of a special kind

Definition 4 (see [16]) A Menger space (119883F Δ) is a tripletwhere (119883F) is a probabilisticmetric space andΔ is a 119905-normsatisfying the following condition

119865119909119910(119905 + 119904) ge Δ (119865

119909119911(119905) 119865119911119910(119904)) (2)

for all 119909 119910 119911 isin 119883 and 119905 119904 ge 0

Definition 5 (see [2]) A pair (119860 119878) of self-mappings definedon a nonempty set 119883 is said to be weakly compatible(or coincidentally commuting) if they commute at theircoincidence points that is if 119860119909 = 119878119909 for some 119909 isin 119883 then119860119878119909 = 119878119860119909

Definition 6 (see [10]) A pair (119860 119878) of self-mappings definedon a nonempty set 119883 is called conversely commuting if forall 119909 isin 119883 119860119878119909 = 119878119860119909 implies 119860119909 = 119878119909

Definition 7 (see [10]) Let 119860 and 119878 be self-mappings of anonempty set 119883 A point 119909 isin 119883 is called commuting pointof 119860 and 119878 if 119860119878119909 = 119878119860119909

Lemma 8 (see [18]) Let (119883F Δ) be a Menger space If thereexists a constant 119896 isin (0 1) such that

119865119909119910(119896119905) ge 119865

119909y (119905) (3)

for all 119905 gt 0 with fixed 119909 119910 isin 119883 then 119909 = 119910

3 Implicit Relations

In 2005 Singh and Jain [19] studied an implicit functionand obtained some fixed point results in framework of fuzzymetric spaces

Let Φ be the set of all real continuous functions 120601

[0 1]4

rarr R nondecreasing in first argument and satisfyingthe following conditions

(120601-1) For 119906 V ge 0 120601(119906 V 119906 V) ge 0 or 120601(119906 V V 119906) ge 0

implies that 119906 ge V(120601-2) 120601(119906 119906 1 1) ge 0 implies that 119906 ge 1


1minus

161199052+ 81199053minus 101199054 Then 120601 isin Φ

Since then Imdad and Ali [20] used the following class ofimplicit functions for the existence of a common fixed pointdue to Popa [21] Many authors proved a number of commonfixed point theorems using the notion of implicit relation ondifferent spaces (see eg [22ndash29])

Let Ψ denote the family of all continuous functions 120595

[0 1]4

rarr R satisfying the following conditions

(120595-1) For every 119906 gt 0 V ge 0 with 120595(119906 V 119906 V) ge 0 or120595(119906 V V 119906) ge 0 we have 119906 gt V

(120595-2) 120595(119906 119906 1 1) lt 0 for all 119906 gt 0


120593(min1199052 1199053 1199054) where 120593 [0 1] rarr [0 1] is a continuous

function such that 120593(119904) gt 119904 for 0 lt 119904 lt 1


119896min1199052 1199053 1199054 where 119896 gt 1


1198961199052minusmin119905

3 1199054 where 119896 gt 0


1198861199052minus 1198871199053minus 1198881199054 where 119886 gt 1 and 119887 119888 ge 0 (119887 119888 = 1)


1198861199052minus 119887(1199053+ 1199054) where 119886 gt 1 and 0 le 119887 lt 1


1minus

119896119905211990531199054 where 119896 gt 1

In 2011 Gopal et al [30] showed that the above-mentioned classes of functions Φ and Ψ are independentclasses

4 Main Results

First we prove a unique common fixed point theorem for twopairs of self-mappings satisfying a class of implicit functionΨ

Theorem 16 Let 119860 119861 119878 and 119879 be four self-mappings ofa Menger space (119883F Δ) where Δ is a continuous 119905-normand the pairs (119860 119878) and (119861 119879) are conversely commutingrespectively and satisfy the following conditions

120595 (119865119860119909119861119910

(119905) 119865119878119909119879119910

(119905) 119865119860119909119878119909

(119905) 119865119861119910119879119910

(119905)) ge 0 (4)

for all 119909 119910 isin 119883 119905 gt 0 and120595 isin Ψ If119860 and 119878 have a commutingpoint and 119861 and 119879 have a commuting point then 119860 119861 119878 and119879 have a unique common fixed point in 119883

Proof Let 119906 be the commuting point of119860 and 119878 Then119860119878119906 =119878119860119906 And let V be the commuting point of 119861 and 119879 Then119861119879V = 119879119861V Since 119860 and 119878 are conversely commuting wehave 119860119906 = 119878119906 Since 119861 and 119879 are conversely commutingwe have 119861V = 119879V Hence 119860119860119906 = 119860119878119906 = 119878119860119906 = 119878119878119906 and119861119861V = 119861119879V = 119879119861V = 119879119879V



120595 (119865119860119906119861V (119905) 119865119878119906119879V (119905) 119865119860119906119878119906 (119905) 119865119861V119879V (119905)) ge 0 (5)

or equivalently

120595 (119865119860119906119861V (119905) 119865119860119906119861V (119905) 1 1) ge 0 (6)




120595 (119865119860119860119906119861V (119905) 119865119878119860119906119879V (119905) 119865119860119860119906119878119860119906 (119905) 119865119861V119879V (119905)) ge 0 (7)

and so

120595 (119865119860119860119906119860119906

(119905) 119865119860119860119906119860119906

(119905) 1 1) ge 0 (8)

Hence for 119865119860119860119906119860119906


120595 (119865119860119906119861119861V (119905) 119865119878119906119879119861V (119905) 119865119860119906119878119906 (119905) 119865119861119861V119879119861V (119905)) ge 0 (9)

or equivalently

120595 (119865119861V119861119861V (119905) 119865119861V119861119861V (119905) 1 1) ge 0 (10)

Thus 119865119860119860119906119860119906



120595 (119865119860119908119861

(119905) 119865119878119908119879

(119905) 119865119860119908119878119908

(119905) 119865119861119879

(119905)) ge 0 (11)

and so

120595 (119865119908(119905) 119865119908(119905) 1 1) ge 0 (12)




119865119909119910(119905) =

119905

119905 +1003816100381610038161003816119909 minus 119910

1003816100381610038161003816

if 119905 gt 0

0 if 119905 = 0(13)


4

rarr R as 120595(1199051 1199052 1199053 1199054) =

1199051minus120593(min119905



119860 (119909) = 2119909 minus 1 if119909 lt 21 if 119909 ge 2

119878 (119909) = 1199092

if 119909 lt 2119909 + 3 if 119909 ge 2

119861 (119909) = 2119909 minus 1 if119909 lt 22 if 119909 ge 2

119879 (119909) = 31199092

minus 2 if 119909 lt 21199092

+ 1 if 119909 ge 2

(14)



119865119860119909119861119910

(119905) ge 120593 (min 119865119878119909119879119910

(119905) 119865119860119909119878119909

(119905) 119865119861119910119879119910

(119905))

(15)


119865119860119909119861119910

(119905) ge 119896 (min 119865119878119909119879119910

(119905) 119865119860119909119878119909

(119905) 119865119861119910119879119910

(119905))

(16)

where 119896 gt 1

119865119860119909119861119910

(119905) ge 119896119865119878119909119879119910

(119905) +min 119865119860119909119878119909

(119905) 119865119861119910119879119910

(119905) (17)

where 119896 gt 0

119865119860119909119861119910

(119905) ge 119886119865119878119909119879119910

(119905) + 119887119865119860119909119878119909

(119905) + 119888119865119861119910119879119910

(119905) (18)

where 119886 gt 1 and 119887 119888 ge 0 (119887 119888 = 1)

119865119860119909119861119910

(119905) ge 119886119865119878119909119879119910

(119905) + 119887 [119865119860119909119878119909

(119905) + 119865119861119910119879119910

(119905)] (19)

where 119886 gt 1 and 0 le 119887 lt 1

119865119860119909119861119910

(119905) ge 119896119865119878119909119879119910

(119905) 119865119860119909119878119909

(119905) 119865119861119910119879119910

(119905) (20)

where 119896 gt 1





120601 (119865119860119909119861119910

(119896119905) 119865119878119909119879119910

(119905) 119865119860119909119878119909

(119905) 119865119861119910119879119910

(119896119905)) ge 0

120601 (119865119860119909119861119910

(119896119905) 119865119878119909119879119910

(119905) 119865119860119909119878119909

(119896119905) 119865119861119910119879119910

(119905)) ge 0

(21)




1 1199052 1199053 1199054) = 18119905

1minus 16119905

2minus 81199053+ 10119905

4besides




120595 (119865119860119909119860119910

(119905) 119865119878119909119878119910

(119905) 119865119860119909119878119909

(119905) 119865119860119910119878119910

(119905)) ge 0 (22)




References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120595 (119865119860119906119861V (119905) 119865119878119906119879V (119905) 119865119860119906119878119906 (119905) 119865119861V119879V (119905)) ge 0 (5)

or equivalently

120595 (119865119860119906119861V (119905) 119865119860119906119861V (119905) 1 1) ge 0 (6)




120595 (119865119860119860119906119861V (119905) 119865119878119860119906119879V (119905) 119865119860119860119906119878119860119906 (119905) 119865119861V119879V (119905)) ge 0 (7)

and so

120595 (119865119860119860119906119860119906

(119905) 119865119860119860119906119860119906

(119905) 1 1) ge 0 (8)

Hence for 119865119860119860119906119860119906


120595 (119865119860119906119861119861V (119905) 119865119878119906119879119861V (119905) 119865119860119906119878119906 (119905) 119865119861119861V119879119861V (119905)) ge 0 (9)

or equivalently

120595 (119865119861V119861119861V (119905) 119865119861V119861119861V (119905) 1 1) ge 0 (10)

Thus 119865119860119860119906119860119906



120595 (119865119860119908119861

(119905) 119865119878119908119879

(119905) 119865119860119908119878119908

(119905) 119865119861119879

(119905)) ge 0 (11)

and so

120595 (119865119908(119905) 119865119908(119905) 1 1) ge 0 (12)




119865119909119910(119905) =

119905

119905 +1003816100381610038161003816119909 minus 119910

1003816100381610038161003816

if 119905 gt 0

0 if 119905 = 0(13)


4

rarr R as 120595(1199051 1199052 1199053 1199054) =

1199051minus120593(min119905



119860 (119909) = 2119909 minus 1 if119909 lt 21 if 119909 ge 2

119878 (119909) = 1199092

if 119909 lt 2119909 + 3 if 119909 ge 2

119861 (119909) = 2119909 minus 1 if119909 lt 22 if 119909 ge 2

119879 (119909) = 31199092

minus 2 if 119909 lt 21199092

+ 1 if 119909 ge 2

(14)



119865119860119909119861119910

(119905) ge 120593 (min 119865119878119909119879119910

(119905) 119865119860119909119878119909

(119905) 119865119861119910119879119910

(119905))

(15)


119865119860119909119861119910

(119905) ge 119896 (min 119865119878119909119879119910

(119905) 119865119860119909119878119909

(119905) 119865119861119910119879119910

(119905))

(16)

where 119896 gt 1

119865119860119909119861119910

(119905) ge 119896119865119878119909119879119910

(119905) +min 119865119860119909119878119909

(119905) 119865119861119910119879119910

(119905) (17)

where 119896 gt 0

119865119860119909119861119910

(119905) ge 119886119865119878119909119879119910

(119905) + 119887119865119860119909119878119909

(119905) + 119888119865119861119910119879119910

(119905) (18)

where 119886 gt 1 and 119887 119888 ge 0 (119887 119888 = 1)

119865119860119909119861119910

(119905) ge 119886119865119878119909119879119910

(119905) + 119887 [119865119860119909119878119909

(119905) + 119865119861119910119879119910

(119905)] (19)

where 119886 gt 1 and 0 le 119887 lt 1

119865119860119909119861119910

(119905) ge 119896119865119878119909119879119910

(119905) 119865119860119909119878119909

(119905) 119865119861119910119879119910

(119905) (20)

where 119896 gt 1





120601 (119865119860119909119861119910

(119896119905) 119865119878119909119879119910

(119905) 119865119860119909119878119909

(119905) 119865119861119910119879119910

(119896119905)) ge 0

120601 (119865119860119909119861119910

(119896119905) 119865119878119909119879119910

(119905) 119865119860119909119878119909

(119896119905) 119865119861119910119879119910

(119905)) ge 0

(21)




1 1199052 1199053 1199054) = 18119905

1minus 16119905

2minus 81199053+ 10119905

4besides




120595 (119865119860119909119860119910

(119905) 119865119878119909119878119910

(119905) 119865119860119909119878119909

(119905) 119865119860119910119878119910

(119905)) ge 0 (22)




References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120601 (119865119860119909119861119910

(119896119905) 119865119878119909119879119910

(119905) 119865119860119909119878119909

(119905) 119865119861119910119879119910

(119896119905)) ge 0

120601 (119865119860119909119861119910

(119896119905) 119865119878119909119879119910

(119905) 119865119860119909119878119909

(119896119905) 119865119861119910119879119910

(119905)) ge 0

(21)




1 1199052 1199053 1199054) = 18119905

1minus 16119905

2minus 81199053+ 10119905

4besides




120595 (119865119860119909119860119910

(119905) 119865119878119909119878119910

(119905) 119865119860119909119878119909

(119905) 119865119860119910119878119910

(119905)) ge 0 (22)




References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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