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# 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 t-representable.

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 non-membership 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 (t-norm) 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 t-norm [tau] on [L.sup.*] is called continuous t-representable if and only if there exist a continuous t-norm * and a continuous t-conorm [??] 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 t-representable 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 3-tuple (X,[M.sub.M,N], [tau]) is said to be an intuitionistic fuzzy metric space if X is an arbitrary (non-empty) set, [tau] is a continuous t-representable 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 t-norms, 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. 87-96, 1986.

[2] G. Deschrijver and E. E. Kerre. On the relationship between some extensions of fuzzy set theory, Fuzzy Sets and Systems, 23, pp. 227-235, 2003.

[3] J.H. Park, Intuitionistic fuzzy metric spaces, Chaos, Solitons and Fractals, 22, pp. 1039-1046, 2004.

Yeol J. Cho (1) and Reza Saadati (2)

(1) Department of Mathematics Education and the RINS, Gyeongsang National University, Chinju 660-701, Korea. E-mail: yjcho@gsnu.ac.kr

(2) Institute for Studies in Applied Mathematics 1, 4th Fajr, Amol 46176-54553, Iran. E-mail: rsaadati@eml.cc
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