INTERNATIONAL JOURNAL OF APPLIED ENGINEERING RESEARCH, DINDIGUL Volume 1, No 4, 2011

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1 Common fixed point theorem for R weakly commutativity of type (Ag) satisfying general contractive type condition Anju Rani, Seema Mehra, Savita Rathee Department of Mathematics, M.D.University, Rohtak toor.anju@yahoo.com ABSTRACT In this paper, we have a common fixed point theorem for a noncompatible pair of mappings satisfying a general contractive condition of integral type in fuzzy metric space which otherwise doesn t ensure a fixed point. We establish a situation in which the common fixed point is a point of discontinuity. Symbols Used for all less than equal to greater than equal to not equal to * t norm ε ϕ ψ α t conorm epsilon belongs to phi shi proper contained alpha Mathematics Subject Classification: 47H1, 54H25. Keywords: Fixed point, R weak commutative of type (Ag), Fuzzy metric space 1. Introduction Fuzzy Theory is one of the best tool to analyze data, when the data under study is an unsupervised one, involving uncertainty coupled with imprecision. Fuzzy set theory was 83

2 formalized by Professor Lofti Zadeh ( 1965 ) at the University of California. After that a lot of work have been done regarding fuzzy set. Especially Deng (1982), Erceg (1979), Kaleva and Seikkala ( 1984) and Kramosil and Michalek ( 1975) have introduced the concept of fuzzy metric space in different ways. Further, Grabiec (1988) followed Kramosil and Michalek (1975) and obtained the fuzzy version of Banach Contraction Principle. In 1975, Kramosil and Michalek introduced the concept of fuzzy metric spaces by generalizing the concept of probabilistic metric space to fuzzy situation. Moreover, George and Verramani (1994) modified the concept of fuzzy metric spaces, introduced by Kramosil and Michalek (1975). George et al. (1994) established a relation between fuzzy metric spaces and metric t spaces as M(x, y, t) =. Every metric space can be made to fuzzy metric space by t + d ( x, y ) the above relation. 2. Materials and methods 2.1 Definition: A fuzzy set A in X is a function with domain X and values in [, 1]. 2.2 Definition: A binary operation * : [, 1] [, 1] [, 1] is a continuous t norm if it satisfies the following conditions: (i) * is associative and commutative, (ii) * is continuous, (iii) a * 1 = a for all a [, 1], (iv) a * b c * d whenever a c and b d, for each a, b, c, d [, 1]. 2.3 Example. Two typical examples of continuous t norm are a * b = ab and a * b = min(a, b). 2.4 Definition. A binary operation : [, 1] [, 1] [, 1] is a continuous t conorm if it satisfies the following conditions : (i) is associative and commutative, (ii) is continuous, (iii) a = a for all a [, 1], (iv) a b c d whenever a c and b d, for each a, b, c, d [, 1]. 831

3 2.5 Example. Two typical examples of continuous t conorm are a b = min(a+b, 1) and a b = max(a, b). 2.6 Definition. A 5 tuple (X,M,N, *, ) is called a intuitionistic fuzzy metric space if X is an arbitrary (non empty) set, * is a continuous t norm, a continuous t conorm and M,N are fuzzy sets on X 2 [, ), satisfying the following conditions : (i)m(x, y, t) + N(x, y, t) 1, (ii) M(x, y, ) = for all x, y X, (iii) M(x, y, t) = 1 for all x, y X and t > if and only if x = y, (iv) M(x, y, t) = M(y, x, t) for all x, y X and t >, (v) M(x, y, t) *M(y, z, s) M(x, z, t + s) for all x, y, z X and t, s >, (vi) for all x, y X, M(x, y,.) : [, ) [, 1] is left continuous. (vii) lim t M(x, y, t) = 1 for all x, y X and t >, (viii) N(x, y, ) = 1 for all x, y X, (ix) N(x, y, t) = for all x, y X and t > if and only if x = y, (x) N(x, y, t) = N(y, x, t) for all x, y X and t >, (xi) N(x, y, t) N(y, z, s) N(x, z, t + s) for all x, y, z X and t, s >, (xii) for all x, y X, N(x, y,.) : [, ) [, 1] is right continuous. (xiii) lim t N(x, y, t) = for all x, y X and t >, Then (M,N) is called an intuitionistic fuzzy metric on X. The functions M(x, y, t) and N(x, y, t) denote the degree of nearness and the degree of non nearness between x and y with respect to t, respectively. 2.7 Remark. Every fuzzy metric space (X,M,.) is an intuitionistic fuzzy metric space of the form (X,M, 1 M, *, ) such that t norm * and t conorm are associated ( Park,24 ), i.e. x y = 1 [(1 x) * (1 y)] for any x, y X. 832

4 2.8 Remark. In intuitionistic fuzzy metric space M(x, y,.) is non decreasing and N(x, y,.) is non increasing for all x, y X. 2.9 Example. (Induced intuitionistic fuzzy metric) Let (X, d) be a metric space. Denote a * b = ab and a b = min(a + b, 1) for all a, b [, 1] and let M d and N d be fuzzy sets on X 2 (, ) defined as follows: M d (x, y, t) = h t n n h t + md ( x, y ) N d (x, y, t) = k t n d ( x, y ) + md ( x, y ) for all h, k,m, n R +. Then (X,M d,n d, *, ) is an intuitionistic fuzzy metric space. 2.1 Definition. A sequence {x n } in a intuitionistic fuzzy metric space (X,M, N, *, ) converges to x if and only if M(x n, x, t) 1 and N(x n, x, t) as n, for each t >. It is called a Cauchy sequence if for each < ε < 1 and t >, there exits n N such that M(x n, x m, t) > 1 ε and N(x n, x m, t) < ε for each n,m n. The intuitionistic fuzzy metric space (X,M,N, *, ) is said to be complete if every Cauchy sequence is convergent. Also, a map f : X X is called continuous at x if {f(x n )} converges to f(x ) for each {x n } converging to x in X. Jungck ( 1986 ) gave the more generalized concept of compatibility than commutativity and weak commutativity in metric space and proved common fixed point theorems. The study of common fixed points of compatible mapping emerged as an area of vigorous research activity every since Jungck introduced the notion of compatibility as follows: A pair of maps f and g:(x,d) (X,d) is said to be compatible if lim n d(fgx n,gfx n ) =, whenever {x n } is a sequence in X such that lim n fx n = lim n gx n = t for some t in X Mishra (1994) introduced the concept of compatible mappings in FM space, akin to concept of compatibility in metric space introduced by Jungck (1986 ), as under: 833

5 Let A and B mappings from a fuzzy metric space (X, M, ) into itself. Then A and B are said to be compatible if lim n M(ABx n,bax n,t) = 1, whenever {x n } is a sequence in X such that lim n Ax n = lim n Bx n = u for some u ε X and for all t >. Clearly, weakly commuting mappings are compatible, but the converse may not be necessarily true. However, the study of common fixed points of noncompatible mappings in metric space has been initiated by Pant ( 1998). Pant (1994) introduced the concept of R weakly commuting of mappings as given below: Two self maps f and g of a metric space (X, d) are called R weakly commuting at a point x ε X if d(fgx,gfx) R d(fx,gx) for some R >. Also f and g are called point wise R weakly commuting on X if given x in X, there exists R > such that d(fgx,gfx) R d(fx,gx). Later on Pathak et.al. (1994) introduced an interesting generalization of R weak commutativity of maps by defining R weak commutativity of type (Ag) as under: Two self maps f and g of a metric space (X,d) are called R weakly commuting of type(ag) if there exists some positive real number R such that d(ffx,gfx) R d(fx,gx) for all x in X. Moreover, such mappings commute at their coincidence points. Now, we introduce the notion of R weak commutativity of type (Ag) in FM spaces akin to metric spaces as follows: Two self maps f and g of an intuitionistic fuzzy metric space (X,M, N, *, ) are called R weakly commuting of type(ag) if there exists some positive real number R such that for all x in X. M(ffx,gfx,t/R) M(fx,gx,t) and N(ffx,gfx,t/R) N(fx,gx,t) for each t > and Moreover, such mappings commute at their coincidence points. 3. Results and Discussions Branciari (22) proved the following fixed point theorem in metric space: 3.1 Theorem. Let (X,d) be a complete metric space, c ε [,1), f: X X a mapping such that, for each x, y ε X, d ( fx, fy ) d ( x, y ) φ(t) dt c φ(t) dt, 834

6 where φ: R + R + is a Lebesgue integrable mapping which is a summable, non negative, and such that, for each >, each x ε X, lim n f n x = z. Now we prove the following result in IFM space: φ(t) dt >. Then f has a unique fixed point z ε X such that, for 3.2 Theorem. Let f and g be noncompatible selfmapping of an intuitionistic fuzzy metric space (X,M, N, *, ) such that (i) f (X ) g (X ),where f (X ) denotes the closure of the range of the mapping f (ii) 1 M ( fx, fy, t ) 1 M ( gx, gy, t ) N ( fx, fy, t ) N ( gx, gy, t ) for each x,y ε X, c ε [,1) where φ: R + R + is a Lebesgue integrable mapping which is a summable, non negative, and such that, for each >, φ(p) dp >. If f and g are R weakly commuting of type(ag) then f and g have a unique common fixed point and the fixed point is a point of discontinuity. Proof Since f and g are noncompatible maps, therefore there exists a sequence {x n } in X such that lim n fx n = lim n gx n = q (iii) for some q in X but lim n M(fgx n,gfx n,t) is either nonunit or nonexistent. Since q ε f (X ) and f (X ) g (X ), therefore, there exists some point u in X such that q = gu. Now, we prove fu = gu, by inequality (ii), we have N ( fx n, fu, t ) 1 M ( fx n, fu, t ) Letting n, we have N ( gx n, gu, t ) 1 M ( gx n, gu, t ) for each x,y ε X, c ε [,1). 835

7 1 M ( gu, fu, t ) 1 M ( gu, gu, t ) N ( gu, fu, t ) N ( gu, gu, t ) which implies fu = gu. Since f and g are R weakly commuting of type(ag), we get M(ffu, gfu,t/r) M(fu,gu,t), N(ffu, gfu,t/r) N(fu,gu,t) i.e., ffu =gfu. Now, our aim is to prove fu= q is common fixed point of f and g, if fu ffu, using (ii) again, we obtain N ( fu, ffu, t ) 1 M ( fu, ffu, t ) N ( gu, gfu, t ) 1 M ( gu, gfu, t ) φ(p) dp = c φ(p) dp = c N ( fu, ffu, t ) 1 M ( fu, ffu, t ) φ(p) dp. which is a contradiction, since c ε [,1), this implies fu = ffu =gfu and therefore, fu is a common fixed point of f and g. Uniqueness follows easily. We now show that f and g are discontinuous at the common fixed point q = fu =gu. If possible, suppose f is continuous, then considering the sequence {x n } of (iii) we get lim n ffx n = fq = q and lim n fgx n = fq. R weak commutative of type(ag) implies that M(ffx n, gfx n,t/r) M(fx n,gx n,t), N(ffx n, gfx n,t/r) N(fx n,gx n,t). On letting limit as n, lim n gfx n = fq = q. This, in turn, yields lim n M(fgx n, gfx n,t) = M(fq,fq,t) = 1, lim n N(fgx n, gfx n,t) = N(fq,fq,t) =, which contradicts the fact that lim n M(fgx n, gfx n,t), lim n N(fgx n, gfx n,t) are either nonunit, nonzero or nonexistent for the sequence {x n } of (iii). Hence f is discontinuous at the fixed point. Next, suppose that g is continuous. Then for the sequence {x n } of (iii), we get lim n gfx n = gq = q and lim n ggx n = gq = q. In view of this, we have N ( fq, fgx n, t ) N ( gq, ggx n, t ) Proceeding limit as n, we have φ(p) dp. 1 M ( fq, fgx n, t ) 1 M ( gq, ggx n, t ) 836

8 lim n fgx n = fq = gq. But lim n gfx n = gq. Therefore, it contradicts the fact that lim n M(fgx n,gfx n,t/r) is either nonunit or nonexistent. Thus f and g are discontinuous at their common fixed point. Hence the theorem. To illustrate the theorem we give an example : 3.3 Example Let X=[3,22] and d be usual metric on X. Let f, g: X X be defined by > 7 ; fx = 7 if 3 < x 7, g3 = 3 ; gx = 1 if 3 < x 7 ; gx = (x+2)/3 if x > 7 and ψ, φ: [, ) [, ), where ψ(t) = (t+1) t+1 1 and φ(t) = t M(x,y, t) = for all t >. t + d(x,y) which gives, fx ={3} {7}, gx =[3,8] {1} and f (X ) g (X ) fx = 3 if x = 3 or x ψ (t). Also let us define. Clearly f and g are noncompatible, since for {x n = 7+1/n, n 1} in X, lim n fx n =3, lim n gx n = 3, lim n fgx n = 7 and lim n gfx n = 3. Hence f and g are noncompatible. Moreover, f and g are discontinuous at the common fixed point x = 3. Thus, f and g satisfy all the conditions of the theorem and have a unique common fixed point at x = Conclusion We prove a common fixed point theorem for non compatible R weakly commuting of type (A g ) mappings in an intuitionistic fuzzy metric space satisfying integral type inequality. Acknowledgement The research of the 1st author on this paper is supported by the grant of the Rajiv Gandhi National fellowship Scheme under University Grant Commission. 5. References 1. A.Branciari, (22), A fixed point theorem for mappings satisfying a general contractive condition of integral type, International journal of Mathematics and Mathematical Sciences, 29(9), pp A. George and P. Veeramani, (1994), On some results in fuzzy metric space, Fuzzy Sets and Systems, 64, pp

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