A common fixed point theorem in intuitionistic fuzzy metric spaces.
Abstract
In this paper, we consider an intuitionistic fuzzy metric space and prove a common fixed point theorem in this space. AMS subject classification: 47H10, 54E50 Keywords: Intuitionistic fuzzy metric spaces, completeness, fixed point theorem. 1. Introduction and Preliminaries In this section, using the idea of intuitionistic fuzzy metric spaces introduced by Park [3] we define the new notion of intuitionistic fuzzy metric spaces with the help of the notion of continuous trepresentable. Lemma 1.1. ([2]) Consider the set [L.sup.*] and operation [[less than or equal to].sub.[L.sup.*] defined by: [L.sup.*] = {([x.sub.1], [x.sub.2]) : ([x.sub.1], [x.sub.2]) [member of] [[0, 1].sup.2] and [x.sub.1] + [x.sub.2] [less than or equal to] 1}, ([x.sub.1], [x.sub.2]) [[less than or equal to].sub.[L.sup.*]] ([y.sub.1], [y.sub.2]) [left and right arrow] [x.sub.1] [less than or equal to] [y.sub.1] and [x.sub.2] [greater than or equal to] [y.sub.2], for every ([x.sub.1], [x.sub.2]), ([y.sub.1], [y.sub.2]) [member of] [L.sup.*]. Then ([L.sup.*], [[less than or equal to].sub.[L.sup.*]]) is a complete lattice. Definition 1.2. ([1]) An intuitionistic fuzzy set [A.sub.[zeta],[eta]] in a universe U is an object [A.sub.[zeta],[eta]] = {([[zeta].sub.A](u), [[eta].sub.A](u))u [member of] U}, where, for all u [member of] U, [[zeta].sub.A](u) [member of] [0, 1] and [[eta].sub.A](u) [member of] [0, 1] are called the membership degree and the nonmembership degree, respectively, of u in [A.sub.[zeta],[eta]], and furthermore they satisfy [[zeta].sub.A](u) + [[eta].sub.A](u) [less than or equal to] 1. Definition 1.3. For every [z.sub.[alpha]] = ([x.sub.[alpha]], [y.sub.[alpha]]) [member of] [L.sup.*] we define [??]([z.sub.[alpha]]) = (sup([x.sub.[alpha]]), inf([y.sub.[alpha]])). Since [z.sub.[alpha]] [member of] [L.sup.*] then [x.sub.[alpha]] + [y.sub.[alpha]] [less than or equal to] 1 so sup([x.sub.[alpha]]) + inf([y.sub.[alpha]]) [less than or equal to] sup([x.sub.[alpha]] + [y.sub.[alpha]]) [less than or equal to] 1, i.e. [??] ([z.sub.[alpha]]) [member of] [L.sup.*]. We denote its units by [0.sub.[L.sub.*]] = (0, 1) and [1.sub.[L.sub.*]] = (1, 0). Classically, a triangular norm * = T on [0, 1] is defined as an increasing, commutative, associative mapping T : [[0, 1].sup.2] [right arrow] [0, 1] satisfying T (1, x) = 1 x = x, for all x [member of] [0, 1]. A triangular conorm S = [??] is defined as an increasing, commutative, associative mapping S : [[0, 1].sup.2] [right arrow] [0, 1] satisfying S(0, x) = 0 [??] x = x, for all x [member of] [0, 1]. Using the lattice ([L.sup.*], [[less than or equal to].sub.[L.sup.*]]) these definitions can be straightforwardly extended. Definition 1.4. ([2]) A triangular norm (tnorm) on [L.sup.*] is a mapping [tau] : [([L.sup.*]).sup.2] [right arrow] [L.sup.*] satisfying the following conditions: ([for all] x [member of] [L.sup.*])([tau] (x, [1.sub.[L.sup.*]]) = x), (boundary condition) ([for all] (x, y) [member of] [([L.sup.*]).sup.2])([tau] (x, y) = [tau] (y, x)), (commutativity) ([for all] (x, y, z) [member of] [([L.sup.*]).sup.3])([tau] (x, [tau] (y, z)) = [tau] ([tau] (x, y), z)), (associativity) ([for all] (x, x', y, y') [member of] [([L.sup.*]).sup.4])(x [[less than or equal to].sub.[L.sup.*]] x' and y [[less than or equal to].sub.[L.sup.*]] y' [??] [tau] (x, y) [[less than or equal to].sub.[L.sup.*]] [tau] (x', y')). (monotonicity). Definition 1.5. ([2]) A continuous tnorm [tau] on [L.sup.*] is called continuous trepresentable if and only if there exist a continuous tnorm * and a continuous tconorm [??] on [0, 1] such that, for all x = ([x.sub.1], [x.sub.2]), y = ([y.sub.1], [y.sub.2]) [member of] [L.sup.*], [tau] (x, y) = ([x.sub.1] * [y.sub.1], [x.sub.2] [??] [y.sub.2]). We say the continuous trepresentable is natural and write [[tau].sub.n] whenever [[tau].sub.n] (a, b) = [[tau].sub.n] (c, d) and a [[less than or equal to].sub.[L.sup.*]] c implies b [[greater than or equal to].sub.[L.sup.*]] d. Definition 1.6. A negator on [L.sup.*] is any decreasing mapping N : [L.sup.*] [right arrow] [L.sup.*] satisfying N([0.sub.[L.sup.*]]) = [1.sub.[L.sup.*]] and N([1.sub.[L.sup.*]]) = [0.sub.[L.sup.*]]. If N(N(x)) = x, for all x [member of] L, then N is called an involutive negator. Definition 1.7. LetM,N are fuzzy sets from [X.sub.2] x (0, +[infinity]) to [0, 1]such that M(x, y, t)+ N(x, y, t) [less than or equal to] 1 for all x, y [member of] X and t > 0. The 3tuple (X,[M.sub.M,N], [tau]) is said to be an intuitionistic fuzzy metric space if X is an arbitrary (nonempty) set, [tau] is a continuous trepresentable and [M.sub.M,N] is a mapping [X.sub.2] x(0,+[infinity]) [right arrow] [L.sup.*] (an intuitionistic fuzzy set, see Definition 1.2) satisfying the following conditions for every x, y [member of] X and t, s > 0: (a) [M.sub.M,N](x, y, t) > [L.sup.*] [0.sub.[L.sup.*]]; (b) [M.sub.M,N](x, y, t) = [1.sub.[L.sup.*]] if and only if x = y; (c) [M.sub.M,N](x, y, t) = [M.sub.M,N](y, x, t); (d) [M.sub.M,N](x, y, t + s) [[greater than or equal to].sub.[L.sup.*]] [tau] ([M.sub.M,N](x, z, t),[M.sub.M,N](z, y, s)); (e) [M.sub.M,N](x, y, x) : (0,[infinity]) [right arrow] [L.sup.*] is continuous. In this case [M.sub.M,N] is called an intuitionistic fuzzy metric. Here, [M.sub.M,N](x, y, t) = (M(x, y, t),N(x, y, t)). Example 1.8. Let (X, d) be a metric space. Denote [tau] (a, b) = ([a.sub.1] [b.sub.1], min([a.sub.2] + [b.sub.2], 1)) for all a = ([a.sub.1], [a.sub.2]) and b = ([b.sub.1], [b.sub.2]) [member of] [L.sup.*] and let M and N be fuzzy sets on [X.sub.2] x (0,[infinity]) defined as follows: [M.sub.M,N](x, y, t) = (M(x, y, t),N(x, y, t)) = ([ht.sup.n]/[ht.sup.n] + md(x, y), md(x, y)/[ht.sup.n] + md(x, y)), for all t, h, m, n [member of] [R.sup.+]. Then (X, [M.sub.M,N], [tau]) is an intuitionistic fuzzy metric space. Lemma 1.9. Let (X, [M.sub.M,N], T) be an intuitionistic fuzzy metric space and define [E.sub.[lambda]], [M.sub.M,N] : [X.sub.2] [right arrow] [R.sup.+] [union] {0} by [E.sub.[[lambda]], [M.sub.M,N] (x, y) = inf{t > 0 : [M.sub.M,N](x, y, t) > [L.sup.*] N([lambda])} for each [lambda] [member of] L \ {[0.sub.[L.sup.*]], [1.sub.[L.sup.*]]} and x, y [member of] X; here, N is an involutive negator. Then we have (i) For any [mu] [member of] L \ {[0.sub.[L.sup.*]], [1.sub.[L.sup.*]]} there exists [lambda] [member of] L {[0.sub.[L.sup.*]], [1.sub.[L.sup.*]]} such that [E.sub.[mu]], [M.sub.M,N] (x, z) [less than or equal to] [E.sub.[lambda]], [M.sub.M,N] (x, y) + [E.sub.[lambda]] [M.sub.M,N] (y, z) for any x, y, z [member of] X; (ii) The sequence [{[x.sub.n]}.sub.n[member of]N] is convergent with respect to intuitionistic fuzzy metric [M.sub.M,N] if and only if [E.sub.[lambda]],[M.sub.M,N] ([x.sub.n], x) [right arrow] 0. Also the sequence {[x.sub.n]} is a Cauchy sequence with respect to intuitionistic fuzzy metric [M.sub.M,N] if and only if it is a Cauchy sequence with [E.sub.[lambda]], [M.sub.M,N]. Proof. For (i), by the continuity of tnorms, for every [mu] [member of] L \ {[0.sub.[L.sup.*]], [1.sub.[L.sup.*]]}, we can find a [lambda] [member of] L\{[0.sub.[L.sup.*]], [1.sub.[L.sup.*]]} such that [tau] (N([lambda]),N([lambda])) [[greater than or equal to].sub.[L.sup.*]] N([mu]). By Definition 1.7 (c), we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for every [delta] > 0, which implies that [E.sub.[mu]], [M.sub.M,N] (x, z) [less than or equal to] [E.sub.[lambda]], [M.sub.M,N] (x, y) + [E.sub.[lambda]], [M.sub.M,N] (y, z) + 2[delta]. Since [delta] > 0 was arbitrary, we have [E.sub.[mu]], [M.sub.M,N] (x, z) [less than or equal to] [E.sub.[lambda]], [M.sub.M,N] (x, y) + [E.sub.[lambda]], [M.sub.M,N] (y, z). For (ii), we have [M.sub.M,N]([x.sub.n], x, [eta]) > [L.sup.*] N([lambda]) [left and right arrow] [E.sub.[lambda]], [M.sub.M,N] ([x.sub.n],x) < [eta] for every [eta] > 0. 2. The Main Results Theorem 2.1. Let {[A.sub.n]} be a sequence of mappings Ai of a complete intuitionistic fuzzy metric space (X, [M.sub.M,N], [tau]) into itself such that, for any two mappings [A.sub.i], [A.sub.j], [M.sub.M,N]([A.sup.m.sub.i] (x), [A.sup.m.sub.j] (y), [[alpha].sub.i,j]t) [[greater than or equal to].sub.[L.sup.*]] [M.sub.M,N](x, y, t) for some m; here 0 < [[alpha].sub.i,j] < k < 1 for i, j = 1, 2, ... , x, y [member of] X and t > 0. Then the sequence {[A.sub.n]} has a unique common fixed point in X. Proof. Let [x.sub.0] be an arbitrary point in X and define a sequence {[x.sub.n]} in X by [x.sub.1] = [A.sup.m.sub.1] ([x.sub.0]), [x.sub.2] = [A.sup.m.sub.2]([x.sub.1]),.... Then we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] and so on. By induction, we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] which implies [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for every [lambda] [member of] L \ {[0.sub.[L.sub.*]], [1.sub.[L.sub.*]]}. Now, we show that {[x.sub.n]} is a Cauchy sequence. For every [mu] [member of] L \ {[0.sub.[L.sub.*]], [1.sub.[L.sub.*]]}, there exists [lambda] [member of] L \ {[0.sub.[L.sub.*]], [1.sub.[L.sub.*]]} such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] as m, n [right arrow] [infinity]. Since X is left complete, there is x [member of] X such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. Now we prove that x is a periodic point of [A.sub.i] for any i = 1, 2,...,. Notice, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] as n [right arrow] [infinity]. Thus [M.sub.M,N](x, [A.sup.m.sub.i](x), t) = [1.sub.[L.sub.*]] and we get [A.sup.m.sub.i] (x) = x. To show uniqueness, assume that y [not equal to] x is another periodic point of [A.sub.i]. Then we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] as n [right arrow [infinity]. Therefore, for every t > 0, we have M(x, y, t) = [1.sub.L], i.e., x=y. Also [A.sub.i](x) = [A.sub.i]([A.sup.m.sub.i] (x)) = [A.sup.m.sub.i] ([A.sub.i](x)), i.e., [A.sub.i](x) is also a periodic point of [A.sub.i]. Therefore, x = [A.sub.i](x), i.e., x is a unique common fixed periodic point of the mappings [A.sub.n] for n = 1, 2,.... This completes the proof. References [1] K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20, pp. 8796, 1986. [2] G. Deschrijver and E. E. Kerre. On the relationship between some extensions of fuzzy set theory, Fuzzy Sets and Systems, 23, pp. 227235, 2003. [3] J.H. Park, Intuitionistic fuzzy metric spaces, Chaos, Solitons and Fractals, 22, pp. 10391046, 2004. Yeol J. Cho (1) and Reza Saadati (2) (1) Department of Mathematics Education and the RINS, Gyeongsang National University, Chinju 660701, Korea. Email: yjcho@gsnu.ac.kr (2) Institute for Studies in Applied Mathematics 1, 4th Fajr, Amol 4617654553, Iran. Email: rsaadati@eml.cc 

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