Common Fixed Point Theorem in Fuzzy Metric Spaces using weakly compatible maps

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1 IJ Iforatio Egieerig ad Electroic Busiess Published Olie April 2014 i MECS ( DOI: /ijieeb Coo Fixed Poit Theore i Fuzzy Metric Spaces usig weakly copatible aps Saurabh Maro School of Matheatics ad Coputer Applicatios Thapar Uiversity Patiala (Pujab) Eail: sauravaro@hotailco sauravaro@yahooco Abstract The ai of this paper is to prove a coo fixed theore for four appigs uder weakly copatible coditio i fuzzy etric space While provig our results we utilize the idea of weakly copatible aps due to Jugck ad Rhoades Our results substatially geeralize ad iprove a ultitude of relevat coo fixed poit theores of the existig literature i etric as well as fuzzy etric space Idex Ters T-or Fuzzy etric space weakly copatible appigs Coo fixed poit theore I INTRODUCTION I 1965 Zadeh [1] itroduced the cocept of Fuzzy set as a ew way to represet vagueess i our everyday life However whe the ucertaity is due to fuzziess rather tha radoess as soeties i the easureet of a ordiary legth it sees that the cocept of a fuzzy etric space is ore suitable We ca divide the ito followig two groups: The first group ivolves those results i which a fuzzy etric o a set X is treated as a ap where X represets the totality of all fuzzy poits of a set ad satisfy soe axios which are aalogous to the ordiary etric axios Thus i such a approach uerical distaces are set up betwee fuzzy objects O the other had i secod group we keep those results i which the distace betwee objects is fuzzy ad the objects theselves ay or ay ot be fuzzy Kraosil ad Michalek [2] have itroduced the cocept of fuzzy etric spaces i differet ways I 1986 Jugck [3] itroduced the otio of copatible aps for a pair of self appigs However the study of coo fixed poits of o-copatible aps is also very iterestig Jugck ad Rhoades [4] iitiated the study of weakly copatible aps i etric space ad showed that every pair of copatible aps is weakly copatible but reverse is ot true I the literature ay results have bee proved for weakly copatible aps satisfyig soe cotractive coditio i differet settigs such as probabilistic etric spaces [5 6 7]; fuzzy etric spaces [8 9 10] I this paper we prove a coo fixed theore for four appigs uder weakly copatible coditio i fuzzy etric space While provig our results we utilize the idea of weakly copatible aps due to Jugck ad Rhoades [4] Our results substatially geeralize ad iprove a ultitude of relevat coo fixed poit theores of the existig literature i etric as well as fuzzy etric space II PRELIMINARIES Defiitio 21[11] A biary operatio * : [01] [01] [01] is cotiuous t-or if * satisfies the followig coditios: (i) * is coutative ad associative; (ii) * is cotiuous; (iii) a * 1 = a for all a [01]; (iv) a * b c * d wheever a c ad b d for all a b c d [01] Kraosil ad Michalek [2] itroduced the cocept of fuzzy etric spaces as follows: Defiitio 22: [2] The 3-tuple (X M *) is called a fuzzy etric space (shortly FM-space) if X is a arbitrary set * is a cotiuous t-or ad M is a fuzzy set i X 2 [0 ) satisfyig the followig coditios: (FM-1) M(x y 0) = 0 (FM-2) M(x y t) = 1 for all t 0 if ad oly if x = y (FM-3) M(x y t) = M(y x t) (FM-4) M(x y t) * M(y z s) M(x z t + s) (Triagular iequality) (FM-5) M(x y): [0 1) [0 1] is left cotiuous for all x y z X ad st 0 Note that M(x y t) ca be thought of as the degree of earess betwee x ad y with respect to t We ca fuzzify exaples of etric spaces ito fuzzy etric spaces i a atural way: Let (X d) be a etric space Defie a * b = a + b for all a b i [01] Defie M(x y t) = t /(t + d(x y)) for all x y i X ad t > 0The (X M *) is a fuzzy etric space ad this fuzzy etric iduced by a etric d is called the Stadard fuzzy etric Copyright 2014 MECS IJ Iforatio Egieerig ad Electroic Busiess

2 Coo Fixed Poit Theore i Fuzzy Metric Spaces usig weakly copatible aps 65 Cosider M to be a fuzzy etric space with the followig coditio: (FM-6) lit M(x y t) = 1 for all x y i X ad t 0 Defiitio 23[2]: Let ( XM *) be fuzzy etric space The ad (a) a sequece { x } i X is said to be Cauchy sequece if for all t 0 ad p 0 li M ( x x t) 1 p (b) a sequece { x } poit x X if for all t 0 li M ( x x t) 1 i X is said to be coverget to a Defiitio 24[2]: A fuzzy etric space (X M *) is said to be coplete if ad oly if every Cauchy sequece i X is coverget Exaple 21[2]: Let X 1 : N 0 ad let * be the cotiuous t-or ad defied by a* b ab for all ab [01] For each t (0 ) ad x y X defie M by t t 0 M ( x y t) t x - y 0 t 0 Clearly (X M *) is coplete fuzzy etric space Defiitio 25[9 12]: A pair of self-appigs (A S) of a fuzzy etric space (X M *) is said to be coutig if M (ASx SAx t) = 1 for all x i X Defiitio 26[9]: A pair of self-appigs (A S) of a fuzzy etric space (X M *) is said to be weakly coutig if M (ASx SAx t) M (Ax Sx t) for all x i X ad t > 0 I 1994 Mishra et al [10] itroduced the cocept of copatible appig i Fuzzy etric space aki to cocept of copatible appig i etric space as follows: Defiitio 27[10]: A pair of self-appigs (A S) of a fuzzy etric space (X M *) is said to be copatible if li M ( ASx SAx t) 1for all t > 0 wheever {x } is a sequece i X such that li Ax li Sx u for soe u i X Defiitio 28[1]: Let (X M *) be a fuzzy etric space A ad S be self aps o X A poit x i X is called a coicidece poit of A ad S iff Ax = Sx I this case w = Ax = Sx is called a poit of coicidece of A ad S Defiitio 29[4]: A pair of self appigs (A S) of a fuzzy etric space (X M *) is said to be weakly copatible if they coute at the coicidece poits ie if Au = Su for soe u X the ASu = SAu It is easy to see that two copatible aps are weakly copatible but coverse is ot true Lea 21[2]: Let {u } is a sequece i a fuzzy etric space (X M *) If there exists a costat k (01) such that M( u u kt) M( u ) 1 u 1 t = the {u } is a Cauchy sequece i X III MAIN RESULTS Theore 31: Let A B P ad Q be self appigs of a coplete fuzzy etric space ( XM *) satisfyig the followig: (31) for ay x y i X ad for all t 0 there exists k (01) such that M ( Ax By t) M ( Px Qy kt) ax 1 M ( Px Ax t) 2 M ( Qx Bx t) (32) P( X ) B( X ) ad Q( X ) A( X ) (33) if oe of P(X) B(X) Q(X) A(X) is coplete subset of X the (a) P ad A have a coicidece poit (b) Q ad B have a coicidece poit If the pair (P A) ad (Q B) are weakly copatible the A B P ad Q have a uique coo fixed poit i X Proof: As P( X ) B( X ) ad Q( X ) A( X ) so we ca defie sequeces { x } ad { y } i X such that y2 1 Px2 Bx2 1 y22 Qx2 1 Ax2 2 By (31) Copyright 2014 MECS IJ Iforatio Egieerig ad Electroic Busiess

3 66 Coo Fixed Poit Theore i Fuzzy Metric Spaces usig weakly copatible aps M ( Ax2 Bx2 M ( Px2 Qx2 1 kt) ax 1 M ( Px2 Ax2 t) 2 M ( Qx2 Bx2 t) M ( y2 y2 M ( y2 1 y22 kt) ax 1 M ( y2 1 y2 t) 2 M ( y2 1 y2 t) M ( y2 y2 M ( y2 1 y22 kt) ax M ( y2 1 y2 t) M ( y y kt) M ( y y t) M ( Ax2 Bx2 2 M ( Px2 Qx2 1 kt) ax M ( Px2 Ax2 t) M ( Qx2 Bx2 t) M ( y2 y2 M ( y2 1 y22 kt) ax 1 M ( y2 1 y2 t) 2 M ( y2 1 y2 t) M ( y2 y2 M ( y2 1 y22 kt) ax M ( y2 1 y2 t) M ( y y kt) M ( y y t) Siilarly M( y22 y23 kt) M( y2 1 y22 t) Therefore i geeral M( y y kt) M( y y t) 1 1 Hece by Lea 21 { y } is Cauchy sequece i X By copleteess of X { y } coverges to soe poit z i X Therefore subsequece s { y 2 }{ y } 2 1 { } coverges to poit z ie y2 2 li Bx li Px 21 2 liqx li Ax z Now suppose A(X) is coplete therefore let 1 w A z the Aw = z Now cosider M ( Aw Bx2 M ( Pw Qx2 1 kt) ax 1 M ( Pw Aw t) 2 M ( Qw Bw t) M ( z y2 M ( Pw y22 kt) ax 1 M ( Pw z t) 2 M ( Qw Bw t) M ( z z t) M ( Pw z kt) ax 1 M ( Pw z t) 2 M ( Qw Bw t) 1 M ( Pw z t) M ( Pw z kt) ax 1 2 M ( Qw Bw t) M ( Pw z kt) 1 This gives Pw = z = Aw Therefore w is coicidece poit of P ad A Sice P( X ) B( X ) therefore z Pw P( X ) B( X ) 1 this gives z B( X ) let v B z ie Bv = z By (31) M ( y2 z t) M ( y2 1 Qv kt) ax 1 M ( y2 1 y2 t) 2 M ( y2 1 y2 t) M ( z z t) M ( z Qv kt) ax 1 M ( z z t) 2 M ( z z t) M ( z Qv kt) 1 This gives Qv = z = Bv So v is coicidece poit of Q ad B Sice the pair (P A) is weakly copatible therefore P ad Q coute at coicidece poit ie PAw = APw this gives Pz = Az ad as (Q B) is weakly copatible therefore QBv = BQv this gives Qz = Bz Now we will show that Pz = z By (31) we have Copyright 2014 MECS IJ Iforatio Egieerig ad Electroic Busiess

4 Coo Fixed Poit Theore i Fuzzy Metric Spaces usig weakly copatible aps 67 M ( Az Bx2 M ( Pz Qx2 1 kt) ax 1 M ( Pz Az t) 2 M ( Qz Bz t) M ( Az y2 M ( Pz y22 kt) ax 1 M ( Az Az t) 2 M ( Bz Bz t) M ( Az z t) M ( Pz z kt) ax 1 M ( Pz z kt) 1 The is gives Pz = z = Az Siilarly we prove that Qz = z By (31) M ( Ax2 Bz t) M ( Px2 Qz kt) ax 1 M ( Px2 Ax2 t) 2 M ( Qx2 Bx2 t) M ( y2 Bz t) M ( y2 1 Qz kt) ax 1 M ( y2 1 y2 1 t) 2 M ( y2 1 y2 t) M ( z Bz t) M ( z Qz kt) ax 1 M ( z z t) 2 M ( z z t) M ( z Bz t) M ( z Qz kt) ax 1 M ( z Qz kt) 1 This gives Qz = z = Bz Therefore z is a coo fixed poit of P A Q ad B For Uiqueess let w be aother fixed poit of P A Q ad B the by (31) we have M ( Az Bw t) M ( Pz Qw kt) ax 1 M ( Pz Az t) 2 M ( Qz Bz t) M ( z w t) M ( z w kt) ax 1 M ( z z t) 2 M ( z z t) M ( z w kt) 1 this gives z = w Hece z is uique coo fixed poit of P A Q ad B By choosig P A Q ad B suitably oe ca derive corollaries ivolvig two or ore appigs As a saple we deduce the followig atural result for a pair of selfappigs by settig P = Q i above theore: Corollary 31: Let A B ad P be self appigs of a coplete fuzzy etric space ( XM *) satisfyig the followig: (34) for ay x y i X ad for all t 0 there exists k (01) such that M ( Ax By t) M ( Px Py kt) ax 1 M ( Px Ax t) 2 M ( Px Bx t) (35) P( X ) B( X ) ad P( X ) A( X ) (36) if oe of P(X) B(X) A(X) is coplete subset of X the (a) P ad A have a coicidece poit (b) P ad B have a coicidece poit If the pair (P A) ad (P B) are weakly copatible the A B ad P have a uique coo fixed poit i X By takig A = B = I i theore 31 we get Corollary 32 : Let P ad Q be self appigs of a coplete fuzzy etric space ( XM *) satisfyig the followig: (37) for ay x y i X ad for all t 0 there exists k (01) such that M ( x y t) M ( Px Qy kt) ax 1 M ( Px x t) 2 M ( Qx x t) (38) if oe of P(X) Q(X) is coplete subset of X If the pair (P Q) is weakly copatible the P ad Q have a uique coo fixed poit i X Defiitio 31 [6] Two failies of self-appigs Ai i 1 ad Bj are said to be pairwise coutig if j 1 (a) A i A j = A j A i i j {123 } (b) B i B j = B j B i i j {123 } (c) A i B j = B j A i i {123 } j {123 } As a applicatio of Theore 31 we prove a coo fixed poit theore for four fiite failies of appigs o fuzzy etric spaces While provig our result we Copyright 2014 MECS IJ Iforatio Egieerig ad Electroic Busiess

5 68 Coo Fixed Poit Theore i Fuzzy Metric Spaces usig weakly copatible aps utilize Defiitio 31 which is a atural extesio of coutativity coditio to two fiite failies Theore 32: Let A A A 1 2 P1 P2 P p ad 1 2 q B1 B2 B Q Q Q be four fiite failies of self appigs of a coplete fuzzy etric space ( XM *) such that A A1 A2 A B B 1 B2 B P P1 P2 P p ad Q Q1 Q2 Q satisfyig the q coditios (31) (32) (33) ad (39) the pairs of B Q coute failies Ai Pk ad r t pairwise The the pairs ( AP ) ad ( BQ ) have a poit of coicidece each Moreover A P rr 1 q p i i1 k k1 B adqt t 1 have a uique coo fixed poit Proof: By usig (39) we first show that AP= PA as AP = (A 1 A 2 A )(P 1 P 2 P p ) = (A 1 A 2 A 1 )(A P 1 P 2 P p ) = (A 1 A 2 A 1 )(P 1 P 2 P p A ) = (A 1 A 2 A 2 )(A 1 P 1 P 2 P p A ) = (A 1 A 2 A 2 )(P 1 P 2 P p A 1 A ) = =A 1 (P 1 P 2 P p A 2 A ) = (P 1 P 2 P p )( A 1 A 2 A ) = PA Siilarly oe ca prove that BQ = QB Ad hece obviously the pairs (A P) ad (B Q) are weakly copatible Now usig Theore 31 we coclude that A B P ad Q have a uique coo fixed poit i X say z Now oe eeds to prove that z reais the fixed poit of all the copoet appigs For this cosider A(A i z) = ((A 1 A 2 A )A i )z = (A 1 A 2 A -1 )(A A i )z = (A 1 A 2 A -1 )(A i A )z = (A 1 A 2 A -2 )(A 1 A i A )z = (A 1 A 2 A -2 )(A i A 1 A )z = A 1 (A i A 2 A )z = (A 1 A i )( A 2 A )z = (A i A 1 )( A 2 A )z = A i (A 1 A 2 A )z = A i Az = A i z Siilarly oe ca prove that ad A(P k z) = P k (Az) = P k z P(P k z) = P k (Pz) = P k z P(A i z)= A i (Pz) = A i z B(B r z) = B r (Bz) = B r z B(Q t z) = Q t (Bz) = Q t z Q(Q t z) = Q t (Qz) = Q t z Q(B r z) = B r (Qz) = B r z which show that (for all i r k ad t) A i z ad P k z are other fixed poit of the pair (A P) whereas B r z ad Q t z are other fixed poits of the pair (B Q) As A B P ad Q have a uique coo fixed poit so we get z = A i z = P k z = B r z = Q t z for all i = 1 2 k = 1 2 p r = 1 2 t = 1 2 q which shows that z is a uique coo fixed poit of p Ai P i1 k k 1 rr 1 B adq 1 Reark 31: Theore 32 is a slight but partial geeralizatio of Theore 31 as the coutativity requireets i this theore are slightly stroger as copared to Theore 31 IV t q t CONCLUSION I this paper we prove a coo fixed theore for four appigs uder weakly copatible coditio i fuzzy etric space While provig our results we utilize the idea of weakly copatible aps Our results substatially geeralize ad iprove a ultitude of relevat coo fixed poit theores of the existig literature i etric as well as fuzzy etric space As a applicatio of Theore 31 we prove a coo fixed poit theore for four fiite failies of appigs o fuzzy etric spaces (Theore 32) REFERENCES [1] LA Zadeh Fuzzy sets Ifor ad Cotrol 8 (1965) [2] I Kraosil ad J Michalek Fuzzy etric ad statistical spaces Kyberetica 11 (1975) [3] G Jugck Copatible appigs ad coo fixed poits Iterat J Math Math Sci 9 (1986) [4] G Jugck ad BE Rhoades Fixed poits for set valued fuctios without cotiuity Idia J Pure Appl Math 29 (1998) [5] J X Fag ad Y Gao Coo fixed poit theores uder strict cotractive coditios i Meger spaces Noliear Aalysis 70 (1) (2009) [6] M Idad J Ali ad M Taveer Coicidece ad coo fixed poit theores for oliear cotractios i Meger PM-spaces Chaos Solitos & Fractals 42 (2009) [7] M Idad M Taveer ad M Hasa Soe coo fixed poit theores i Meger PM-spaces Fixed Poit Theory ad Applicatios Volue 2010 Article ID pages [8] S Maro SS Bhatia ad S Kuar Coo fixed poit theores i fuzzy etric spaces Aals of Fuzzy Matheatics ad Iforatics 3(1)(2012) [9] S Maro S Kuar ad SS Bhatia A Addedu to Sub-Copatibility ad Fixed Poit theore i Fuzzy Metric Space Iteratioal Joural of Fuzzy Matheatics ad Systes 2(2) (2012) [10] S N Mishra N Shara ad S L Sigh Coo fixed poits of aps o fuzzy etric spaces It J Math Math Sci 17 (1994) [11] B Schweizer ad A Sklar Probabilistic Metric Spaces North Hollad Asterda 1983 Copyright 2014 MECS IJ Iforatio Egieerig ad Electroic Busiess

6 Coo Fixed Poit Theore i Fuzzy Metric Spaces usig weakly copatible aps 69 [12] G Jugck Coutig appigs ad fixed poits Aerica Matheatical Mothly 83(1976) Authors Profiles Saurabh Maro is the Research Scholar i School of Matheatics ad Coputer Applicatios Thapar Uiversity Patiala (Pujab) Idia He copleted MSc (Matheatics) i year 2004 fro Pujab Uiversity Chadigarh He is pursuig PhD uder the kid supervisio of Dr SS Bhatia ad Dr Sajay Kuar His areas of research iclude Fixed poit theore i various abstract spaces like Meger spaces Probabilistic Metric spaces Fuzzy Metric spaces Ituitioistic Fuzzy Metric spaces G- Metric spaces ad its applicatios He has published ay research papers i atioal / iteratioal papers till ow Soe papers are ready to be published How to cite this paper: Saurabh Maro"Coo Fixed Poit Theore i Fuzzy Metric Spaces usig weakly copatible aps" IJIEEB vol6 o2 pp DOI: /ijieeb Copyright 2014 MECS IJ Iforatio Egieerig ad Electroic Busiess