Common Fixed Point Theorems for Four Weakly Compatible Self- Mappings in Fuzzy Metric Space Using (JCLR) Property

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1 IOSR Joural of Mathematics (IOSR-JM) e-issn: , p-issn: X. Volume, Issue 3 Ver. I (May - Ju. 05), PP Commo Fixed Poit Theorems for Four Weakly Compatible Self- Mappigs i Fuzzy Metric Space Usig (JCLR) Property N. Appa Rao, Dr. V. Dharmaiah, Departmet of Mathematics Hyderabad (TS) Departmet of Dr. B.R. Ambedkar Ope Uiversity Osmaia Uiversity Mathematics Hyderabad (TS) apparaoemaala968@gmail.com Abstract: I this paper we prove a commo fixed poit theorem for four weakly compatible self-mappigs i fuzzy metric space by usig (JCLR) property. A example is give which shows the validity of mai theorem. We also exted the result to two fiite families of self-mappigs with pairwise commutig. Key Words: Fuzzy Metric Space, Weakly Compatible Mappigs, E.A Property, (CLR) Property ad (JCLR) Property. I. Itroductio I 965, Zadeh [33] itroduced the cocept of fuzzy set. I the last two decades tremedous developmet has take place i the study of fuzzy set theory. Fuzzy set theory has applicatios i applied scieces such as eural etwork, stability theory, mathematical programmig, modelig theory, image processig, cotrol theory, commuicatio, egieerig scieces, medical scieces etc. I 975, Kramosil ad Michalek [8] itroduced the cocept of fuzzy metric space. George ad Veeramai [] modified the cocept of fuzzy metric space itroduced by Kramosil ad Michalek [8], which opeed o aveue for further developmet of aalysis i fuzzy metric spaces. I 00, Aamri ad El-Moutawakil [] defied the otio of E.A property for self-mappigs which cotais the class of o-compatible mappigs i metric spaces. E.A property replaces the completeess coditio by closedess coditio of the rage ad relaxes the completeess of the whole space, cotiuity of oe or more mappigs ad cotaimet of the rage of oe mappig ito the rage of the other. A umber theorems were proved for self-mappigs satisfyig E.A property [or CLR s property]. May authors have proved commo fixed poit theorems i fuzzy metric spaces for differet cotractive coditios. For details refer to [3, 5-6, 7, 0,, 4, 5, 6, 7, 8, 3, 3]. II. Prelimiaries. Fuzzy Metric Space A fuzzy metric space is a triple (X, M, T ) where X is a oempty set, T is a cotiuous t-orm ad M is a fuzzy set o X ( 0, ) ad the followig coditios are satisfied for all x, y X ad t, s > 0: (FM-) M (x, y, t)> 0; (FM-) M (x, y, t )= x = y; (FM-3) M (x, y, t )= M (y, x, t); (FM-4) M (x, y, ): (0, ) [0, ] is cotiuous; (FM-5) M (x, z, t + s ) T (M (x, y, t ), M (y, z, s)).. Weakly Compatible Mappigs Two self-mappigs f ad g of a o-empty set X are said to be weakly compatible (or coicidetally commutig) if they commute at their coicidece poits, i.e., if fz = gz for some z X, the fgz = gfz..3 E.A. Property Two self-mappigs f ad g of a fuzzy metric space (X, M, T ) are said to satisfy E.A. property, if there exists a sequece {x } i X such that lim fx = lim gx = u for some u X..4 (CLR) Property DOI: / Page

2 Let f ad g be self-mappigs o a fuzzy metric space (X, M, T ). The the pair ( f, g ) is said to satisfy CLRg property (commo limit i the rage of g property) if lim fx = lim gx = gx for some x X..5 Icreasig Fuctio Let Φ be class of all mappigs ϕ :[0, ] [0, ] satisfyig the followig coditios: ( ϕ) ϕ is cotiuous ad odecreasig o [0, ]; (ϕ ) ϕ(x) > x for all x (0, )..6 Lemma Let (X, M, T ) be a fuzzy metric space. If there exists k ( 0, ) such that M (x, y, kt ) M (x, y, t) for all x, y X ad t > 0, the x = y. III. Mai Result I this paper, we first itroduce the otio of the joit commo limit i the rage property of two pairs of self-mappigs. 3. Defiitio Let (X, M, T ) be a fuzzy metric space ad f, g, p, q : X X be self-mappigs. The pairs ( f, q) ad (p, g ) are said to satisfy the joit commo limit i the rage of q ad g property (shortly, (JCLRqg) property) if there exist two sequeces {x } ad {y } i X such that lim fx = lim qx = lim py = lim gy for some u X. = qu = gu, -- () 3. Remark If p = f, q = g ad {x } ={y } of (CLRg) property. i (), the we get the defiitio Throughout this sectio, Φ deotes the set of all cotiuous ad icreasig fuctios φ :[ 0, ] 5 [ 0, ] i ay coordiate ad φ (t, t, t, t, t ) > t for all t [0, ). Followig are examples of some fuctios φ Φ: ) φ ( x, x, x 3, x 4, x 5 ) = ( mi{ x i }) h, for some 0 < h <. ) φ (x, x, x 3, x 4, x 5 ) = x h for some 0 < h <. 3) φ ( x, x, x 3, x 4, x 5 )=T (T ( T ( T ( x, x ), x 3 ), x 4 ), x 5 ) h, for some 0 < h < ad for all t-orms T such that T (t, t ) = t. Now, we state ad prove our mai result. 3.3 Theorem Let (X, M, T ) be a fuzzy metric space ad f, g, p ad q be mappigs from X ito itself. Further, let the pairs ( f, q) ad (p, g ) are weakly compatible ad there exists a costat k 0, such that (qx, gy, t ), M ( fx, qx, t ), M (py, gy, t), DOI: / Page

3 M ( fx, py, kt) φ -- () M ( fx, gy, α t ), M ( py, qx, t αt) holds for all x, y X, α ( 0, ), t > 0 ad φ Φ. If ( f, q) ad (p, g ) satisfy the (JCLRqg) property, the f, g, p ad q fixed poit i X. have a uique commo Sice the pairs ( f, q) ad (p, g ) satisfy the (JCLRqg) property, there exist sequeces {x } ad {y } i X such that lim fx = lim qx = lim py = lim gy for some u X. Now, we assert that gu = pu. Usig (), with x = x, y = u, α =, we get = qu = gu, M (qx, gu, t ), M ( fx, qx, t ), M (pu, gu, t), M ( fx, pu, kt) φ Takig the limit as, we have M ( fx, gu, t ), M ( pu, qx, t) M ( gu, pu, kt) M (gu, gu, t ), M (gu, gu, t ), M (pu, gu, t), φ. M ( gu, gu, t ), M ( pu, gu, t) Sice φ is icreasig i each of its coordiate ad φ (t, t, t, t, t ) > t for all t [0, ), we get M (gu, pu, kt ) M (gu, pu, t). By Lemma.6, we have gu = pu. Next, we show that fu = gu. Usig (), with x = u, y = y, α =, we get M ( M fu, py, kt) φ ( gu, gy, t ), M ( fu, qu, t ), M M ( fu, gy, t ), M Takig the limit as, we have M (gu, gu, t ), M ( fu, gu, t ), M M ( fu, gu, kt) φ M ( fu, gu, t ), M Sice φ is icreasig i each of its coordiate ad φ (t, t, t [0, ), we get M ( fu, gu, kt ) M ( fu, gu, t). By Lemma.6, we have fu = gu. ( py, gy, t ),. ( py, qu, t) ( gu, gu, t),. ( gu, gu, t) t, t, t ) > t for all DOI: / Page

4 Now, we assume that z = fu = gu = pu = qu. Sice the pair ( f, q) is weakly compatible, fqu = qfu ad the fz = fqu = qfu = qz. Agai sice (p, g ) is weakly compatible, it follows gpu = pgu ad hece gz = gpu = pgu = pz. We show that z = fz. To prove this, usig (), with x = z, y = u, α =, we get M (qz, gu, t ), M ( fz, qz, t ), M (pu, gu, t), M ( fz, pu, kt) φ ad so M ( fz, gu, t ), M ( pu, qz, t) M ( fz, z, t ), M ( fz, fz, t ), M (z, z, t), M ( fz, z, kt) φ. M ( fz, z, t ), M ( z, fz, t) Sice φ is icreasig i each of its coordiate ad φ (t, t, t, t, t ) > t for all t [0, ), M ( fz, z, kt ) M ( fz, z, t), which implies that fz = z. Hece z = fz = qz. Next, we show that z = pz. To prove this, usig (), with x = u, y = z, α =, M ( fu, pz, kt) we get M φ (qu, gz, t ), M ( fu, qu, t ), M (pz, gz, t), ad so M ( fu, gz, t ), M ( pz, qu, t) M ( z, pz, kt) φ. M (z, pz, t ), M (z, z, t ), M (pz, pz, t), M ( z, pz, t ), M ( pz, z, t) Sice φ is icreasig i each of its coordiate ad φ (t, t, t, t, t ) > t for all t [0, ),M (z, pz, kt ) M (pz, pz, t), which implies that z = pz. Hece z = pz = gz. Therefore, we coclude that z = fz = gz = pz = qz. This implies f, g, p ad q have a commo fixed poit z. For uiqueess of commo fixed poit, we let w be aother commo fixed poit of the mappigs f, g, p ad q. O usig () with x = z, y = w, α =, we have M (qz, gw, t ), M ( fz, qz, t ), M (pw, gw, t), M ( fz, pw, kt) φ M ( fz, gw, t ), M ( pw, qz, t) ad the M ( z, w, kt) φ. M (z, w, t ), M (z, z, t ), M (w, w, t), M ( z, w, t ), M ( w, z, t) DOI: / Page

5 Sice φ is icreasig i each of its coordiate ad φ (t, t, t, t, t ) > t for all t [0, ), M (z, w, kt ) M (z, w, t), which implies that z = w. Therefore f, g, p ad q have a uique commo fixed poit. 3.4 Remark From the results, it is asserted that (JCLRqg) property ever requires coditios closedess of the subspace, cotiuity of oe or moremappigs ad cotaimet of rages amogst ivolved mappigs. 3.5 Remark Theorem 3.3 improves ad geeralizes the results of Abbas et al. ([], Theorem.) ad Kumar ([9], Theorem.3) without ay requiremet of cotaimet amogst rage sets of the ivolved mappigs ad closedess of the uderlyig subspace. 3.6 Remark Sice the coditio of t-orm with T (t, t ) = t for all t [ 0, ] is replaced by arbitrary cotiuous t-orm, Theorem 3.3 also improves the result of Cho et al. ([4], Theorem 3.) without ay requiremet of completeess of the whole space, cotiuity of oe or more mappigs ad cotaimet of rages amogst ivolved mappigs. 3.7 Corollary Let (X, M, T ) be a fuzzy metric space ad f, g, p ad q be mappigs from X ito itself. Further, let the pairs ( f, q) ad (p,g ) are weakly compatible ad there exists a costat k 0, such that M ( fx, py, kt) a ( t ) M ( qx, gy, t) + a ( t ) M ( fx, qx, t) + a ( t ) M ( py, gy, t) 3 +a ( t ) M ( fx, gy, α t ) + a ( t ) M ( py, qx, t αt) (3) holds for all x, y X, α ( 0, ), t > 0 ad a : + (0, ] such that i 5 i = a i ( t )=. If ( f, q) ad (p, g ) satisfy the (JCLRqg) property, the f, g, p ad q have a uique commo fixed poit i X. By Theorem 3.3, we defie a ( t ) x+ a ( t ) x + a ( t )x φ ( x, x, x, x, x )= a ( t ) x + a ( t )x the the result follows. 3.8 Remark Corollary 3.7 improves the result of Cho et al. ([4], Theorem 3.) without ay requiremet of completeess of the whole space, cotiuity of oe or more mappigs ad cotaimet of rages amogst DOI: / Page

6 ivolve mappigs while the coditio of t-orm T (t, t )= t for all t [ 0, ] is replaced by a arbitrary cotiuous t-orm. 3.9 Corollary Let (X, M, T ) be a fuzzy metric space ad f, g be mappigs from X ito itself. Further, let the pair ( f, g ) be weakly compatible ad there exists a costat k 0, such that M (gx, gy, t ), M ( fx, gx, t ), M ( fy, gy, t), M ( fx, fy, kt) φ -- (4) M ( fx, gy, α t ), M ( fy, gx, t αt) holds for all x, y X, α ( 0, ), t > 0 ad φ Φ. If ( f, g ) satisfies the (CLRg) property, the f ad g have a uique commo fixed poit i X. Take p = f ad q = g i Theorem 3.3, the we get the result. Our ext theorem is proved for a pair of weakly compatible self- mappigs i fuzzy metric space (X, M, T ) usig E.A property uder additioal coditio of closedess of a subspace. 3.0 Theorem Let (X, M, T ) be a fuzzy metric space. Further, let the pair ( f, g ) of self-mappigs be weakly compatible satisfyig iequality (4) of Corollary 3.9. If f ad g satisfy E.A property ad the rage of g is a closed subspace of X, the f ad g have a uique commo fixed poit ix. Sice the pair ( f, g ) satisfies E.A property, there exists a sequece {x } i X such that lim fx = lim gx = z for some z X. It follows, from g (X ) beig a closed subspace of X that there exists u X such that z = gu. Therefore f ad g satisfy the (CLRg) property. From Corollary 3.9, the result follows. I what follows, we preset some illustrative examples which demostrate the validity of the hypothesis ad degree of utility of our results. 3. Example Let X =[, 9) with the metric d defied by d (x, y ) = x y ad for each t [ 0, ] defie t M ( x, y, t)= t + x y,if t > 0 0,if t = 0 for all x, y X. Clearly (X, M, T ) is a fuzzy metric space with t-orm defied by T (a, b )= mi{a, b} for all a, b [0, ]. Cosider a fuctio φ( x, x, x 3, x 4, x 5 ) = ( mi{ x i }). The φ :[ 0, ] 5 [ 0, ] defied by we have M ( fx, fy, t ) φ (x, x, x 3, x 4, x 5 ). Defie the self-mappigs f ad g o X by DOI: / Page

7 , if x { } (3, 9) fx = 5, if x (, 3]., if x = ad gx =, if x (, 3] x +, if x ( 3, 9). Takig { x }= 3 + or {x }={}, it is clear that the pair ( f, g ) satisfies the (CLRg) property sice It is oted that f lim fx = lim gx = = g ( ) X. ( X ) = {, 5, 0 = g } / [ ) { } ( X ). Thus, all the coditios of Corollary 3.9 are satisfied for a fixed costat k 0, ad is a uique commo fixed poit of the self-mappigs f ad g. Also, all the ivolved mappigs are eve discotiuous at their uique commo fixed poit. Here, it may be poited out that g (X ) is ot a closed subspace of X. 3. Example I the settig of Example 3., replace the mappig g by the followig, besides retaiig the rest:, gx = 0, x + if x = if x (, 3] if x ( 3, 9), DOI: / Page

8 Takig { x }= 3 + satisfies the E.A property sice lim fx = lim gx = X. or {x } ={}, it is clear that the pair ( f, g ) It is oted that f (X ) = {, 5} / [, 0]= g (X ). Thus, all the coditios of Corollary 3.9 are satisfied ad is a uique commo fixed poit of the mappigs f ad g. Notice that all the ivolved mappigs are eve discotiuous at their uique commo fixed poit. Here, it is worth otig that g (X ) is a closed subspace of X. Our ext theorem exteds Corollary 3.9 to two fiite families of self-mappigs which are pairwise commutig. 3.3 Theorem Let { f i } m ad {g j } be two fiite families of self-mappigs i a i= j= fuzzy metric space (X, M, T ) such that f = f f... f m ad g = g g...g which satisfy the iequality (4) of Corollary 3.9. If the pair ( f, g ) satisfies (CLRg) property, the f ad g have a uique poit of coicidece. Moreover, { f i } m ad { g j } have a uique commo fixed poit i= j= provided the pair of families ({ f i },{g j}) commute pairwise, where i {,,, m} ad j {,,, }. The proof of this theorem ca be completed o the lies of Theorem 3. cotaied i Imdad et al. [7], hece details are avoided. Puttig f = f =... = f m = f ad g = g =... = g = g i Theorem 3.3, we get followigs result: 3.4 Corollary Let f ad g be two self-mappigs of a fuzzy metric space (X, M, T ). Further, let the pair (f m, g ) satisfies (CLRg) property. The there exists a costat k 0, such that M ( f m x, f y, kt) φ M (g x, g y, t ), M (f m x, g x, t ), M (f m y, g y, t), m m M ( f x, g y, α t ), M ( f y, g x, t αt) holds for all x, y X, α ( 0, ), t > 0, φ Φ ad m ad are fixed positive itegers, the f ad g have a uique commo fixed poit provided the pairs ( f m, g ) commute pairwise. DOI: / Page

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