Common Fixed Point Theorems for Generalized Fuzzy Contraction Mapping in Fuzzy Metric Spaces

Transcription

1 Volume 119 No , ISSN: (on-line version) url: ijpam.eu Common Fixed Point Theorems for Generalized Fuzzy Contraction Mapping in Fuzzy Metric Spaces 1 R. Pandiselvi, 2 M. Jeyaraman and 3 A. Urvasi 1 Department of Mathematics, The Madura College, Madurai, Tamil Nadu. rpselvi@gmail.com 2 PG and Research Department of Mathematics, Raja Doraisingam Govt. Arts College, Sivaganga, Tamil Nadu, India. jeya.math@gmail.com 3 PG and Research Department of Mathematics, Raja Doraisingam Govt. Arts College, Sivaganga, Tamil Nadu, India. a.urvasi92@gmail.com Abstract In this paper, we investigate common fixed point theorems for contraction mapping in fuzzy metric space introduced by Gregori and Sapena. Key Words:common fixed point, generalized fuzzy metric spaces, generalized contraction mappings. AMS Mathematics Subject Classification (2010): 47H10, 54H

2 1. Introduction The concept of fuzzy sets was introduced by Zadeh [12] in Since then, to use this concept in topology and analysis many authors have expansively developed the theory of fuzzy sets and applications. The concept of Fixed point theorems in fuzzy mathematics is emerging with vigorous hope and vital trust. It appears that Kramosil and Michalek s study of fuzzy metric spaces paves a way for very soothing machinery to develop fixed point theorems for contractive type maps. Kramosil and Michalek [6] introduced the concept of fuzzy metric space and modified by George and Veeramani [2]. Many authors [5, 8] have proved fixed point theorems in fuzzy metric space. Sedghi and Shobe [10] introduced D* - metric space as a probable modification of the definition of D - metric introduced by Dhage [1], and prove some basic properties in D* - metric spaces. Using D* - metric concepts, Sedghi and Shobe define the M-fuzzy metric space and proved a common fixed point theorem in it. In this paper, we introduce fixed point theorems for generalized contraction mapping in M - fuzzy metric spaces. Following the results of Gregori and Sapena[5] we give a new common fixed point theorem in the two different types of completeness and by using their recent definition of contractive mapping in M-fuzzy metric spaces. 2. Preliminaries Definition 2.1. A binary operation : [0, 1] x [0,1] satisfies the following conditions (i) (ii) is associative and commutative, is continuous, (iii) a 1 = a, for all a [0, 1], is a continuous t norm if it (iv) a b c d whenever a c and b d, for each a, b, c, d [0, 1]. Examples for continuous t norm are a b = min {a, b}and a b = ab Definition 2.2. A 3 tuple (X,, ) is called - fuzzy metric space (generalized fuzzy metric space) if X is an arbitrary non empty set, is a continuous t norm, and is a fuzzy set on X 3 x (0, ), satisfying the following conditions: for each x, y, z, a X and t, s > 0. (FM 1) (x, y, z, t)> 0, 14644

3 (FM 2) (x, y, z, t) = 1 if x = y= z, (FM 3) (x, y, z, t) = ( {x, y, z}, t), where is a permutation function, (FM 4) (x, y, a, t) * (a, z, z, s) (x, y, z, t +s), (FM 5) (x, y, z,.): (0, ) [0, 1] is continuous, (FM 6) (x, y, z, t) = 1. Definition 2.3. A sequence {x n } in X is said to be convergent to a point x in X (denoted by x n x), if (x n, x, x, t) 1, for all t > 0. Definition 2.4. Let (X,, *) be a fuzzy metric space. (i) A sequence {x n } is called G-Cauchy if (x n+p, x n, x n, t) = 1 for each t > 0 and p N. The fuzzy metric space (X,, ) is called G-complete if every G- Cauchy sequence is convergent. (ii) A sequence{x n } in a fuzzy metric space (X,, ) is a Cauchy sequence,if for each and each t > 0 there exists n 0 N such that M(x n, x m,x m, t) > 1 - for all n, m n 0. The fuzzy metric space (X, called complete if every Cauchy sequence is convergent. Proposition 2.5. (a) A sequence {x n } in the metric space X is contractive in (X, d)iff {x n } is fuzzy contractivein the induced fuzzy metric space (X, D, ). (b) The standard fuzzy metric space (X, D, min) is complete iff the metric space (X, d) is complete. (c) If the sequence {x n } is fuzzy contractive in (X,, ) then it is G-Cauchy. 3. Main Results Definition 3.1. Let (X,, ) be a fuzzy metric space. (a) We call the mapping T : X X is fuzzy contractive mapping, if there exists λ (0,1) such that 1, for each x, y, z X and t > 0., ) is 14645

4 (b) A sequence {x n } is called fuzzy contractive if there exists λ (0, 1)such that for every t > 0, n N. Theorem 3.2 Let (X,, ) be a G-complete fuzzy metric space endowed with minimum t-norm and {T α } α J be a family of self mappings of X. If there exist fixed β, Jsuch that for each α J 1 1 1M,,, 1+ 21M,,, 1+ 31M,,, M,,, M,,, 1 (3.2.1) for each x, y, z X, t > 0 and for some <1. Then all T α have a unique fixed point and if 0 < 1, 0 + <1 then at this point each T α is continuous. Proof: Let J and x X be arbitrary. Consider a sequence, defined inductively by x = x n, y = x n+1, z = = x n+1 and x 2n+1 = T α x 2n, x 2n+2 = T β x 2n+1, x 2n+2 = T γ x 2n+1 for all n 0. From (3.2.1) we get =. (3.2.2) Since = max 14646

5 +.(3.2.3) Combine equations (3.2.1) and (3.2.2) we get (1- - ) ( + + ) Hence, λ, Where, by the assumption, λ= get that belongs to (0, 1). Similarly, we λ. So {x n }is fuzzy contractive, since X is G-complete, {x n } converges to u for some u X. From (3.2.1) we have = M 2, 2 +1, 2 +1, M, 2 +1, 2 +1, M(,, 2 +1, 2 1. Taking the limitas infinity we obtain. Thus ( u, T u, T u, t) = 1, hence, T γu = T βu = u. Now we show that u is a fixed point of all {T α J}. Let α J. From (3.2.1) and Remark, we have = 4 1M,,, 1+ 51M(,,, )

6 + ( + ). Hence = = u, since α is arbitrary all { } αϵ J have a common point. Suppose that v is also a fixed point of. Similar to above, v is a common fixed point of all { } αϵ J. From (3.2.1) we get =. Thus u is a unique common fixed point of all { } αϵ J. it remains to show each T α is continuous at u. Let {y n }be a sequence in X such that y n u as n. From (3.2.1) we have = 4 1M(,,, ) 1+ 51M(,,,, ) 1,(3.2.4) and similar to (3.2.3) we have max. (3.2.5) Combine (3.2.4) and (3.2.5) we deduce + + max,(3.2.6) for all t > 0, n N. So by (3.2.6), we have inf ( sup ( sup (, (3.2.7) for all t > 0, thus 14648

7 (3.2.8) ( = ( = L, exists, for all t > 0, and then L equals 1, since in opposite case, applying (3.2.6) -(3.2.8), one can easily concluded that + 1, contrary to assumotion. Thus T α is continuous at a fixed point. The mapping in the preceeding theorem is called generalized contraction mapping. Note that every fuzzy contractive mapping satisfies the condition (3.2.1). Theorem 3.3. Let (X,, ) be a complete non-archimedean fuzzy metric space endowed with minimum t-norm and {T α } α J be a family of self mappings of X. If there exist a fixed β J such that for each α J for each x, y X, t > 0 and for some <1. Then all T α have a unique fixed point and at this point each T α is continuous. Proof: The proof is very similar to Theorem (3.2) Instead of the equation (3.2.3) we have =max. Proceed as the as the proof of the Theorem (3.2) then we conclude sequence {x n } is fuzzy contractive, {x n } converges to u for some u X. Proceed as the proof of Theorem (3.2). The following provide a converse to Theorem (3.2). Theorem 3.4 Let (X, M, ) be a G-complete fuzzy metric space endowed with minimum t-norm. The following property is equivalent to G- completeness of X: 14649

8 If Y is any non empty closed subset of X and T:Y Y is any generalized contraction mapping then T has a fixed point in Y. Proof: The sufficient condition follows from Theorem (3.2) Suppose now that the property holds, but (X,, ) is not complete. Then there exists a Cauchy sequence {x n } in X which does not converge. We may assume that (x n, x m, x m, x) < 1for all m n and for some t > o. For any x X define r(x) = inf. Clearly for all x X we have r(x) > 0, as {x n } has not convergent subsequence. Let = = = 1/8. we choose a subsequence { } of {x n } as follows. We define inductively a subsequence of positive integer greater than and such that -1 r( ) for all i,k i n, n 1. This can done, as {x n } is a cauchy sequence. Now, we define = for all n. Then for any n > m 0 we have r( ) = =. Thus T is a general contraction mapping on Y = { }. Clearly, Yis closed and T has not a fixed point in Y. Thus we get a contradiction. 4. Conclusion In this paper, a theorem on the existence of a common fixed point is proved which characterizes G-completeness of fuzzy metric spaces

9 References [1] Dhage B.C, Generalized metric spaces and mappings with fixed point, Bull. Calcutta Math, Soc., 84(4) (1992), [2] George. A and Veeramani. P, On some results in fuzzy metric spaces, Fuzzy sets and systems. 64 (1994), [3] Grabiee.M, Fixed points in fuzzy metric spaces, Fuzzy Sets and Systems 27 (1998), [4] Gregori. V, and Romaguera.S, On completion of fuzzy metric spaces, Fuzzy Sets and Systems, Theme, Fuzzy intervals, 130 (3) (2002), [5] Gregori V. and Sapena. A., On fixed point theorem in fuzzy metric spaces, Fuzzy Sets and Systems 125 (2002), [6] Kramosil. I and Michalek. J, Fuzzy metric and statistical spaces, Kybernetica 11 (1975), [7] Mihet.D, Fuzzy ψ-contractive mappings in non- Archimedean fuzzy metric spaces, Fuzzy Sets and Systems 159 (6) (2008), [8] Mihet. D, A Banch Contraction theorem in fuzzy metric spaces, Fuzzy Sets and Systems 144 (2004), [9] Park.J.H, Park J S and Kwun.Y.C, Fixed Point in -fuzzy metric spaces, Optimization and Decision Making 7 (4) (2008), [10] Sedghi.S and Shobe.N, Fixed point theorem in -fuzzy metric spaces with property (E), Advances in Fuzzy Mathematics 1(1) (2006), [11] Veerapandi.T, Jeyaraman. M and Paul Raj Joseph.J, Some fixed point and coincident point theorem in generalized -fuzzy metric space, Int. Journal of Math. Analysis 3 (2009), [12] Zadeh L.A, Fuzzy sets, Inform. and Control 8 (1965),

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