A Common Fixed Point Theorem in Intuitionistic Fuzzy. Metric Space by Using Sub-Compatible Maps

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1 It. J. Cotemp. Math. Scieces, Vol. 5, 2010, o. 55, A Commo Fixed Poit Theorem i Ituitioistic Fuzzy Metric Space by Usig Sub-Compatible Maps Saurabh Maro*, H. Bouharjera** ad Shivdeep Sigh*** *School of Mathematics ad Computer Applicatios, Thapar Uiversit Patiala(Pujab) **Uiversite europeee de Bretage, Frace *sauravmaro@yahoo.com **hakima.bouhadjera@uiv-brest.fr ***shivdhidsa@gmail.com Abstract: I this paper, we itroduce the ew cocepts of subcompatibility ad subsequecial cotiuity which are respectively weaker tha occasioally weak compatibility ad reciprocal cotiuity. With them, we establish a commo fixed poit theorem for four maps. Our results exted ad ituitioistic fuzzify the results of [4]. Mathematics Subject Classificatio: 47H10, 54H25 Keywords: Ituitioistic fuzzy metric space, Subcompatibility ad Subsequecial cotiuit commo fixed poit theorem 1. Itroductio Ataassov [3] itroduced ad studied the cocept of ituitioistic fuzzy sets as a geeralizatio of fuzzy sets [14]. I 2004, Park [9] defied the otio of ituitioistic fuzzy metric space with the help of cotiuous t-orms ad cotiuous t-coorms. Recetl i 2006, Alaca et al.[1] usig the idea of ituitioistic fuzzy sets, defied the otio of ituitioistic fuzzy metric space with the help of cotiuous t-orm ad cotiuous t-coorms as a geeralizatio of fuzzy metric space due to Kramosil ad Michalek [6]. Further, Alaca et al.[1] proved Ituitioistic fuzzy Baach ad Ituitioistic

2 2700 S. Maro, H. Bouharjera ad S. Sigh fuzzy Edelstei cotractio theorems, with the differet defiitio of Cauchy sequeces ad completeess tha the oes give i [9]. I this paper, we itroduce the ew cocepts of subcompatibility ad subsequecial cotiuity which are respectively weaker tha occasioally weak compatibility ad reciprocal cotiuity. With them, we establish a commo fixed poit theorem for four maps. Our results exted ad ituitioistic fuzzify the results of [4]. 2. Prelimiaries Defiitio 2.1 [10]. A biary operatio :[0,1] [0,1] [0,1] is a cotiuous t -orm if is satisfyig the followig coditios: (a) is commutative ad associative; (b) is cotiuous; (c) a1 a for all a[0, 1]; (d) a b c d wheever a c ad b d for all a, b, c, d [0,1]. Examples of t -orm are a b mi{ a, b} ad a b ab. Defiitio 2.2 [10]. A biary operatio :[0,1] [0,1] [0,1] is a cotiuous t -coorm if is satisfyig the followig coditios: (a) is commutative ad associative; (b) is cotiuous; (c) a0 a for all a[0, 1]; (d) ab cd wheever a c ad b d for all a, b, c, d [0,1]. Examples of t -coorm are ab max{ a, b} ad a b mi{ 1, a b}. Alaca et al. [1] defied the otio of ituitioistic fuzzy metric space as follows: Defiitio 2.3[1]. A 5-tuple ( X, M, N,, ) is said to be a IFM-space if X is a arbitrary set, is a cotiuous t -orm, is a cotiuous t -coorm ad M, N are fuzzy 2 sets o X [0, ) satisfyig the followig coditios: for all z X ad t, s 0, (i) M ( N( 1; (ii) M ( 0) 0; (iii) M ( 1 if ad oly if x y; (iv) M ( M ( ; (v) M ( M ( z, s) M ( z, t s); (vi) M ( ) :[0, ) [0, 1] is left cotiuous; (vii) limt M ( 1; (viii) N( 0) 1; (ix) N( 0 if ad oly if x y;

3 Commo fixed poit theorem 2701 (x) N( N( ; (xi) N( N( z, s) N( z, t s); (xii) N( ) :[0, ) [0, 1] is right cotiuous; (xiii) limt N( 0. The ( M, N) is called a ituitioistic fuzzy metric o X. The fuctios M ( ad N( deote the degree of earess ad the degree of o-earess betwee x ad y with respect to t, respectively. Remark 2.1. Every fuzzy metric space ( X, M, ) is a IFM-space of the form ( X, M, 1 M,, ) such that t -orm ad t -coorm are associated, i.e. x y 1 ((1 x) (1 y)) for ay y X. Example 2.1. Let ( X, d) be a metric space. Defie t -orm a b mi{ a, b} ad t - coorm a b max{ a, b} ) ad for all y X ad t 0, t M ( d( y) d, N ( t d( y) t d( y). d The ( X, M, N,, ) is a IFM-space ad the ituitioistic fuzzy metric ( M, N) iduced by the metric d is ofte referred to as the stadard ituitioistic fuzzy metric. Remark 2.2. I IFM-space ( X, M, N,, ), M ( ) is o-decreasig ad N( ) is o-icreasig for all y X. Defiitio 2.4 [1]. Let ( X, M, N,, ) be a IFM-space. The (i) a sequece { x } i X is said to be Cauchy sequece if for all t 0 ad p 0, lim M ( x, x, 1, lim N ( x, x, 0. p p (ii) a sequece x } i X is said to be coverget to a poit x X if for all t 0, lim M ( x {, 1, lim N ( x, 0. Sice ad are cotiuous, the limit is uiquely determied from (v) ad (xi) respectively. Defiitio 2.5 [1]. A IFM-space ( X, M, N,, ) is said to be complete if ad oly if every Cauchy sequece i X is coverget. 1 Example 2.2. Let X = { : = 1, 2, 3,... } {0} ad let * be the cotiuous t-orm ad be the cotiuous t-coorm defied by a * b = ab ad a b = mi{1, a + b} 0,1 t 0, ad y X, defie (M, N) by respectivel for all a, b. For each

4 2702 S. Maro, H. Bouharjera ad S. Sigh t, t > 0 t+ x-y x-y, t > 0 t+ x-y. M( = ad N( = 0, t =0 1, t =0 Clearl (X, M, N, *, ) is complete ituitioistic fuzzy metric space. The followig defiitio of weakly commutig mappigs i ituitioistic fuzzy metric space is give o the lies of Sessa [11]. Defiitio 2.6[11]. Let A ad S be maps from a ituitioistic fuzzy metric space ( X, M, N,, ) ito itself. The maps A ad S are said to be weakly commutig if M ( ASz, SAz, M ( Az, Sz, ad N ( ASz, SAz, N( Az, Sz, for all z X ad t 0. Defiitio 2.7[13]. Let A ad S be maps from a IFM-space ( X, M, N,, ) ito itself. The maps A ad S are said to be compatible if for all t 0, lim M ( AS SA 1 ad lim N ( AS SA 0, wheever { x } is a sequece i X such that lim Ax lim Sx z for some z X. Defiitio 2.8[7]: Two mappigs A ad S of a ituitioistic fuzzy metric space ( X, M, N,, ) will be called reciprocally cotiuous if ASu Az ad SAu Sz, wheever { u } is a sequece such that Au, Su z for some z i X. If A ad S are both cotiuous, the they are obviously reciprocally cotiuous but coverse is ot true. Moreover, i the settig of commo fixed poit theorems for compatible pair of mappigs satisfyig cotractive coditios, cotiuity of oe of the mappigs A ad S implies their reciprocal cotiuity but ot coversely. Defiitio 2.9: Let (X, M, N, *, ) be a ituitioistic fuzzy metric space. A ad S be self maps 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 2.10: A pair of self mappigs (A, S ) of a ituitioistic fuzzy metric space (X, M, N, *, ) is said to be weakly compatible if they commute at the coicidece poits i.e., if Au = Su for some u i X, the ASu = SAu. It is easy to see that two compatible maps are weakly compatible but coverse is ot true. Defiitio 2.11[2]: Two self mappigs A ad S of a ituitioistic fuzzy metric space (X, M, N, *, ) are said to be occasioally weakly compatible (owc) iff there is a poit x i X which is coicidece poit of A ad S at which A ad S commute. I this paper, we weake the above otio by itroducig a ew cocept called subcompatibility just as defied by H. Bouhadjera[4] i metric space, as follows:

5 Commo fixed poit theorem 2703 Defiitio 2.12: Let (X, M, N,*, ) be a ituitioistic fuzzy metric space. Self maps A ad S o X are said to be subcompatible iff there exists a sequece {x } i X such that lim Ax lim Sx z, z X ad satisfy lim M ( ASx, SAx, 1, lim N( ASx, SAx, 0. Obviousl two owc maps are subcompatible, however the coverse is ot true i geeral. The example below shows that there exist subcompatible maps which are ot owc. Example 2.3: Let X [0, ). For each t (0, ) ad yx, defie (M, N) by t, t > 0, t+ x-y x-y, t > 0, M( = ad N( = t+ x-y 0 t =0 1 t =0 Defie A ad S as follows: 2 x 2 if x [0,4] (9, ) A( x) x, S( x). x 12 if x (4,9] 1 Let {x } be a sequece i X defied by x 2 for = 1, 2, 3, The, lim Ax lim Sx 4, 4 X ad ASx 16, SAx 16 whe. Thus, lim M ( ASx, SAx, 1, lim N( ASx, SAx, 0. i.e. A ad S are subcompatible. O the other had, we have Ax = Sx iff x = 2 ad AS(2) SA(2), hece A ad S are ot owc. Now, our secod objective is to itroduce subsequetial cotiuity i ituitioistic fuzzy metric space which weake the cocept of reciprocal cotiuity which was itroduced by Pat[8] just as itroduced by H. Bouhadjera[4] i metric space, as follows: Defiitio 2.13: Let (X, M,N,*, ) be a ituitioistic fuzzy metric space. Self maps A ad S o X are said to be subsequetially cotiuous iff there exist a sequece {x } i X such that lim Ax lim Sx t, t X ad satisfy lim ASx At, lim SAx St. Clearl if A ad S are cotiuous or reciprocally cotiuous the they are obviously subsequetially cotiuous. The ext example shows that there exist subsequetial cotiuous pairs of maps which are either cotiuous or reciprocally cotiuous. Example 2.4: Let X [0, ). For each t (0, ) ad yx, defie (M, N) by M( = t, t > 0, t+ x-y 0 t =0 x-y, t > 0, ad N( = t+ x-y 1 t =0

6 2704 S. Maro, H. Bouharjera ad S. Sigh Defie A ad S as follows: 1 x if x [0,1] 1 x if x [0,1) A( x), S( x) 2x 1 if x (1, ) 3x 2 if x [1, ). Clearly A ad S are discotiuous at x = 1. 1 Let {x } be a sequece i X defied by x for = 1, 2, 3, The, lim Ax lim Sx 1,1 X ad ASx 2 A(1), SAx 1 S(1) whe, therefore, A ad S are subsequetial cotiuous. 1 Now, let {x } be a sequece i X defied by x 1 for = 1, 2, 3, The, lim Ax lim Sx 1,1 X ad ASx 1 2 A(1) whe, so A ad S are ot reciprocally cotiuous. Lemma 2.1 Let { u } be a sequece i a ituitioistic fuzzy metric space ( X, M, N,, ). If there exists a costat k (0,1) such that M ( u, u 1, k M ( u 1, u, ad N ( u, u 1, k N ( u 1, u,, for all t>0 ad =1,2,3 The u } is a Cauchy sequece i X. { Lemma 2.2 Let ( X, M, N,, ) be a IFM-space ad for all y X, t 0 a umber k (0,1), M ( k M ( ad N( k N( the x y. ad if for 3. Mai Result Now, we prove ours mai theorem usig defiitio of subcompatible ad subsequetial cotiuous maps as follows: Theorem 3.1: Let A, B, S ad T be four self maps of a Ituitioistic fuzzy metric space (X,M,N,*, ) with cotiuous t-orm * ad cotiuous t-coorm defied by t t t ad (1 (1 (1 t 0,1. If the pairs (A, S) ad (B, T) are subcompatible ad for all subsequetially cotiuous, the (a) A ad S have a coicidece poit; (b) B ad T have a coicidece poit.

7 Commo fixed poit theorem 2705 Further, let for all y i X, k 0,1, t > 0 M ( A B k M ( S T M ( A S M ( B T M ( B S2 M ( A T, (c) N( A B k N( S T N( A S N( B T N( B S 2 N ( A T. The, A, B, S ad T have a uique commo fixed poit. Proof: Sice, the pairs (A, S) ad (B, T) are subcompatible ad subsequetially cotiuous, the, there exists two sequeces {x } ad {y } i X such that lim Ax lim Sx z, z X ad satisfy lim M ( ASx, SAx, M ( Az, Sz, 1,lim N( ASx, SAx, N( Az, Sz, 0; lim By lim Ty z ', z ' X ad which satisfy lim M ( BTy, TBy, M ( Bz ', Tz ', 1;lim N( BTy, TBy, N( Bz ', Tz ', 0. Therefore, Az = Sz ad Bz ' Tz '; that is, z is a coicidece poit of A ad S ad z ' is a coicidece poit of B ad T. Now, we prove that z z '. By iequality (c), we have take x = x ad y = y, M ( Ax, By, k M ( Sx, Ty, M ( Ax, Sx, M ( By, Ty, M ( By, Sx, 2 M ( Ax, Ty,, N( A B k N( S T N( A S N( B T N( B S 2 N( A T. takig the limit as yields M ( z, z ', k M ( z, z ', M ( z, z, M ( z ', z ', M ( z ', z,2 M ( z, z ',, N( z, z ', k N( z, z ', N( z, z, N( z ', z ', N( z ', z, 2 N ( z, z ', M ( z, z ', k M ( z, z ',, N( z, z ', k N( z, z ',. By lemma 2.2, we have z z '. Also, we claim that Az = z. By iequality (c), take x = z ad y = y, we get M ( Az, By, k M ( Sz, Ty, M ( Az, Sz, M ( By, Ty, M ( By, Sz, 2 M ( Az, Ty,, N( Az, B k N( Sz, T N( Az, Sz, N ( B T N( B Sz, 2 N ( Az, T. takig the limit as yields M ( Az, z ', k M ( Az, z ', M ( Az, Az, M ( z ', z ', M ( z ', Az, 2 M ( Az, z ',, N( Az, z ', k N( Az, z ', N( Az, Az, N( z ', z ', N ( z ', Sz,2 N ( Az, z ',. M ( Az, z ', k M ( Az, z ',, N( Az, z ', k N( Az, z ', By lemma 2.2, Az z ' z. Agai, we claim that Bz = z. Usig (c), take x = z ad y = z, we get M ( Az, Bz, k M ( Sz, Tz, M ( Az, Sz, M ( Bz, Tz, M ( Bz, Sz,2 M ( Az, Tz,, N( Az, Bz, k N( Sz, Tz, N( Az, Sz, N( Bz, Tz, N ( Bz, Sz, 2 N ( Az, Tz,. M ( z, Bz, k M ( z, Bz, M ( Az, Az, M ( Bz, Bz, M ( Bz, z, 2 M ( z, Bz,, N( z, Bz, k N( z, Bz, N( Az, Az, N( Bz, Bz, N ( Bz, z,2 N ( z, Bz,. M ( z, Bz, k M ( z, Bz,, N( z, Bz, k N( z, Bz,. By lemma 2.2, z = Bz =Tz.

8 2706 S. Maro, H. Bouharjera ad S. Sigh Therefore, z = Az = Bz = Sz = Tz; that is z is commo fixed poit of A, B, S ad T. For Uiqueess: Suppose that there exist aother fixed poit w of A, B, S ad T. By coditio (c), take x = z, y = w, we have M ( Az, Bw, k M ( Sz, Tw, M ( Az, Sz, M ( Bw, Tw, M ( Bw, Sz,2 M ( Az, Tw,, N( Az, Bw, k N( Sz, Tw, N( Az, Sz, N ( Bw, Tw, N( Bw, Sz, 2 N ( Az, Tw,. M ( z, w, k M ( z, w,, N( z, w, k N( z, w, Hece, by lemma 2.2, z = w. Therefore, uiqueess follows. If we put S = T, i Theorem 3.1, we get the followig result: Corollary 3.2: Let A, B ad S be three self maps of a Ituitioistic fuzzy metric space (X,M,N,*, ) with cotiuous t-orm * ad cotiuous t-coorm defied by t t t ad (1 (1 (1 t 0,1. If the pairs (A, S) ad (B, S) are subcompatible ad for all subsequetially cotiuous, the (d) A ad S have a coicidece poit; (e) B ad S have a coicidece poit. k 0,1, t > 0 Further, let for all y i X, M ( A B k M ( S S M ( A S M ( B S M ( B S2 M ( A S, (f) N( A B k N( S S N( A S N( B S N( B S 2 N ( A S. The, A, B ad S have a uique commo fixed poit. Refereces [1] C. Alaca, D. Turkoglu, ad C. Yildiz, Fixed poits i Ituitioistic fuzzy metric spaces, Chaos, Solitos & Fractals, 29 (2006), [2] M.A. Al-Thagafi, N. Shahzad, Geeralised I-oexpasive self maps ad ivariats approximatios. Acta Math. Si, 24 (5) (2008), [3] K. Ataassov, Ituitioistic Fuzzy sets, Fuzzy sets ad system, 20 (1986) [4] H. Bouhadjera ad C. Godet-Thobie, Commo fixed poit Theorems for pairs of subcompatible maps, arxiv: v1 [math.fa] 17 jue [5] G. Jugck, Commo fixed poits for o cotiuous o self maps o o-metric spaces. Far East J. Math. Sci., 4 (2) (1996), [6] I. Kramosil ad J. Michalek, Fuzzy metric ad Statistical metric spaces, Kyberetica 11 (1975) [7] S. Muralisakar ad G. Kalpaa, Commo fixed poit theorems i ituitioistic fuzzy metric spaces usig reeral cotractive coditio of itegral type, It. J. Cotemp. Math. Scieces, 4 (11) (2009), [8] R.P. Pat, K. Jha, A remark o commo fixed poits of four mappigs i a fuzzy metric space, J. Fuzzy Math., 12(2) (2004),

9 Commo fixed poit theorem 2707 [9] J.H. Park, Ituitioistic fuzzy metric spaces, Chaos, Solitos & Fractals, 22 (2004) [10] B. Schweizer ad A. Sklar, Probabilistic Metric Spaces, North Hollad Amsterdam, [11] S. Sessa, O a weak commutativity coditio of mappigs i fixed poit cosideratios, Publ. Ist. Math. (Beograd)(N. S.), 32 (46) (1982), [12] D. Turkoglu, C. Alaca, Y.J. Cho ad C. Yildiz, Commo fixed poit theorems i ituitioistic fuzzy metric spaces, J. Appl. Math & Computig, 22 (1-2) (2006), [13] D. Turkoglu, C. Alaca ad C. Yildiz, Compatible maps ad Compatible maps of type ( ) ad ( ) i ituitioistic fuzzy metric spaces, Demostratio Math. 39 (3) (2006), [14] L.A. Zadeh, Fuzzy sets, Ifor. ad Cotrol. 8 (1965), Received: Jul 2010