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Fixed points of occasionally weakly compatible mappings in fuzzy metric spaces.

[section] 1. Introduction

In 1965, Zadeh [40] introduced the concept of fuzzy sets. Since then, to use this concept in topology and analysis, many authors have expansively developed the theory of fuzzy sets and applications. For example, Kramosil and Michalek [32], Erceg [21], Deng [20], Kaleva and Seikkala [30], Grabiec [24], Fang [22], George and Veeramani [23], Mishra et al. [33], Subrahmanyam [38], Gregori and Sapena [25] and Singh and Jain [37] have introduced the concept of fuzzy metric spaces in different ways. In applications of fuzzy set theory, the field of engineering has undoubtedly been a leader. All engineering disciplines such as civil engineering, electrical engineering, mechanical engineering, robotics, industrial engineering, computer engineering, nuclear engineering etc. have already been affected to various degrees by the new methodological possibilities opened by fuzzy sets.

In 1998, Jungck and Rhoades [27] introduced the notion of weakly compatible mappings in metric spaces. Singh and Jain [37] formulated the notion of weakly compatible maps in fuzzy metric spaces. This condition has further been weakened by introducing the notion of owc maps by Al-Thagafi and Shahzad [7]. While Khan and Sumitra [31] extended the notion of owc maps in fuzzy metric spaces and proved some common fixed point theorems. It is worth to mention that every pair of weak compatible self-maps is owc but the reverse is not always true. Many authors proved common fixed point theorems for owc maps on various spaces (see [1-15, 17-19, 28, 29, 31, 34, 35, 39]).

In this paper, we prove some common fixed point theorems for owc maps in fuzzy metric spaces. Our results do not require the completeness of the whole space or any subspace, continuity of the involved maps and containment of ranges amongst involved maps.

[section] 2. Preliminaries

Definition 2.1. [36] A triangular norm * (shortly t-norm) is a binary operation on the unit interval [0,1] such that for all a, b, c, d [member of] [0,1] and the following conditions are satisfied:

1. a * 1 = a,

2. a * b = b * a,

3. a * b [less than or equal to] c * d whenever a [less than or equal to] c and b [less than or equal to] d,

4. (a * b) * c = a * (b * c).

Two typical examples of continuous t-norms are a * b = min{a, b} and a * b = ab.

Definition 2.2. [32] A 3-tuple (X, M, *) is said to be a fuzzy metric space if X is an arbitrary set, * is a continuous t-norm and M is a fuzzy set on [X.sup.2] x (0, [infinity]) satisfying the following conditions, for all x, y, z [member of] X, t, s > 0,

1. M(x, y, t) =0,

2. M(x, y, t) = 1 if and only if x = y,

3. M(x, y, t) = M(y, x, t),

4. M(x, z, t + s) [greater than or equal to] M(x, y, t) * M(y, z, s),

5. M(x, y, x) : (0, [infinity]) [right arrow] (0,1] is continuous.

Then M is called a fuzzy metric on X. Then M(x, y, t) denotes the degree of nearness between x and y with respect to t.

Example 2.3. [23] Let (X, d) be a metric space. Denote a * b = a.b (or a * b = min{a, b}) for all a, b [member of] [0,1] and let [M.sub.d] be fuzzy sets on [X.sup.2] x (0, [infinity]) defined as follows:

Md(x, y, t) = t/(t+d{x,y)).

Then (X, [M.sub.d], *) is a fuzzy metric space. We call this fuzzy metric induced by a metric d.

Lemma 2.4. [16,24] For all x, y [member of] X, (X, M, x) is non-decreasing function.

Definition 2.5. [37] Let (X, M, *) be a fuzzy metric space, A and B be self maps of non-empty X. A point x [member of] X is called a coincidence point of A and B if and only if Ax = Bx. In this case w = Ax = Bx is called a point of coincidence of A and B.

Definition 2.6. [37] Two self mappings A and B of a fuzzy metric space (X, M, *) are said to be weakly compatible if they commute at their coincidence points, that is, if Ax = Bx for some x [member of] X then ABx = BAx.

Lemma 2.7. If a fuzzy metric space (X,M, *) satisfies M(x, y, t) = C, for all t > 0 with fixed x, y [member of] X. Then we have C = 1 and x = y.

Lemma 2.8. [26] Let the function [phi](t) satisfy the following condition ([PHI]): [phi](t) : [0, [infinity]) [right arrow] [0, [infinity]) is non-decreasing and [[summation].sup.[infinity].sub.n=1][[phi].sup.n](t) < [infinity] for all t > 0, when [[phi].sup.n](t) denotes the nth iterative function of [phi](t). Then [phi](t) < t for all t > 0.

The following concept due to Al-Thagafi and Shahzad [7-8] is a proper generalization of nontrivial weakly compatible maps which do have a coincidence point. The counterpart of the concept of owc maps in fuzzy metric spaces is as follows:

Definition 2.9. Two self maps A and B of a fuzzy metric space (X, M, *) are owc if and only if there is a point x [member of] X which is a coincidence point of A and B at which A and B commute.

From the following example it is clear that the notion of owc maps is more general than the concept of weakly compatible maps.

Example 2.10. Let (X, M, *) be a fuzzy metric space, where X = [0, [infinity]) and

M (x, y, t) = t/(t+[absolute value of x-y])

for all t > 0 and x, y [member of] X. Define A, B : X [right arrow] X by A(x) = 3x and B(x) = [x.sup.2] for all x [member of] X. Then A(x) = B(x) for x = 0, 3 but AB(0) = BA(0) and AB(3) [not equal to] BA(3). Thus A and B are owc maps but not weakly compatible.

The following lemma is on the lines of Jungck and Rhoades [28].

Lemma 2.11. Let (X, M, *) be a fuzzy metric space, A and B are owc self maps of X. If A and B have a unique point of coincidence, w = Ax = Bx, then w is the unique common fixed point of A and B.

Proof. Since A and B are owc, there exists a point x in X such that Ax = Bx = w and ABx = BAx. Thus, AAx = ABx = BAx, which says that Ax is also a point of coincidence of A and B. Since the point of coincidence w = Ax is unique by hypothesis, BAx = AAx = Ax, and w = Ax is a common fixed point of A and B.

Moreover, if z is any common fixed point of A and B, then z = Az = Bz = w by the uniqueness of the point of coincidence.

[section] 3. Results

First, we prove a common fixed point theorem for four single-valued self maps in fuzzy metric space.

Theorem 3.1. Let A,B,S and T be self maps on fuzzy metric space (X, M, *), where * is a continuous t-norm with a * a [greater than or equal to] a for all a [member of] [0,1]. Further, let the pairs (A, S) and (B, T) are each owc satisfying:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

for all x, y [member of] X and t > 0. Here, the function [phi](t) : [0, [infinity]) [right arrow] [0, [infinity]) is onto, strictly increasing and satisfies condition ([PHI]). Then there exists a unique point w [member of] X such that Aw = Sw = w and a unique point z [member of] X such that Bz = Tz = z. Moreover, z = w, so that there is a unique common fixed point A, B, S and T.

Proof. Since the pairs (A, S) and (B,T) are each owc, there exist points u, v [member of] X such that Au = Su, ASu = SAu and Bv = Tv, BTv = TBv. Now we show that Au = Bv. Putting x = u and y = v in inequality (1), then we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

then we have

M(Au, Bv, < [phi](t)) [greater than or equal to] M(Au, Bv, t).

On the other hand, since M is non-decreasing, we get M(Au, Bv, [phi](t)) [less than or equal to] M(Au, Bv, t). Hence, M(Au, Bv, t) = C for all t > 0. From Lemma 2.7, we conclude that C = 1, that is Au Bv. Therefore, Au = Su = Bv = Tv. Moreover, if there is another point z such that Az Sz. Then using inequality (1) it follows that Az = Sz = Bv = Tv, or Au = Az. Hence w = Au = Su is the unique point of coincidence of A and S. By Lemma 2.11, w is the unique common fixed point of A and S. Similarly, there is a unique point z [member of] X such that z = Bz = Tz. Suppose that w [not equal to] z and taking x = w, y = z in inequality (1), then we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

thus it follows that

M(w, z, [phi](t)) [greater than or equal to] M(w, z, t).

Since M is non-decreasing, we get M(w, z, [phi](t)) [less than or equal to] M(w, z, t). Hence, M(w, z, t) = C for all t > 0. From Lemma 2.7, we conclude that C = 1, that is w = z. Hence w is the unique common fixed point of the self maps A, B, S and T in X.

Now, we give an example which illustrates Theorem 3.1.

Example 3.2. Let X = [0,4] with the metric d defined by d(x, y) = [absolute value of x - y] and for each t [member of] [0,1] define

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

for all x, y [member of] X. Clearly (X, M, *) be a fuzzy metric space, where * is a continuous t-norm with * = min. Define [phi](t) = kt, where k [member of] (0,1) and the self maps A, B, S and T by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Then A, B, S and T satisfy all the conditions of Theorem 3.1. Notice that AS(2) = A(2) = 2 = S(2) = SA(2) and BT(2) = B(2) = 2 = T(2) = TB(2), that is A and S as well as B and T are owc. Hence, 2 is the unique common fixed point of A, B, S and T. This example never requires any condition on containment of ranges amongst involved maps. On the other hand, it is clear to see that the self maps A, B, S and T are discontinuous at 2.

On taking A = B and S = T in Theorem 3.1, then we get the following result:

Corollary 3.3. Let A and S be self maps on fuzzy metric space (X, M, *) where * is a continuous t-norm and a * a [greater than or equal to] a for all a [member of] [0, 1]. Further, let the pair (A, S) is owc satisfying:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

for all x, y [member of] X and t > 0. Here, the function [phi](t) : [0, [infinity]) [right arrow] [0, [infinity]) is onto, strictly increasing and satisfies condition ([PHI]). Then A and S have a unique common fixed point in X.

Now, we extend Theorem 3.1 and Corollary 3.3 to any even number of self-maps in fuzzy metric space.

Theorem 3.4. Let [P.sub.1], [P.sub.2], ..., [P.sub.2n], A and B be self maps on fuzzy metric space (X, M, *), where * is a continuous t-norm with a * a [greater than or equal to] a for all a [member of] [0, 1]. Further, let the pairs (A, [P.sub.1][P.sub.3] ... [P.sub.2n-1]) and (B, [P.sub.2][P.sub.4] ... [P.sub.2n]) are each owc satisfying:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

for all x, y [member of] X and t > 0. Here, the function [phi](t) : [0, [infinity]) [right arrow] [0, [infinity]) is onto, strictly increasing and satisfies condition ([PHI]). Suppose that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

similarly,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Then [P.sub.1],[P.sub.2], ..., [P.sub.2n], A and B have a unique common fixed point in X.

Proof. Since the pairs (A, [P.sub.1][P.sub.3] ... [P.sub.2n-1]) and (B, [P.sub.2][P.sub.4] ... [P.sub.2n]) are each owc then there exist points u, v [member of] X such that Au = [P.sub.1][P.sub.3] ... [P.sub.2n-1]u, A([P.sub.1][P.sub.3] ... [P.sub.2n-1])u = ([P.sub.1][P.sub.3] ... [P.sub.2n-1])Au and Bv = [P.sub.2][P.sub.4] ... [P.sub.2n]v, B([P.sub.2][P.sub.4] ... [P.sub.2n])v = ([P.sub.2][P.sub.4] ... [P.sub.2n])Bv. Now we show that Au = Bv. Taking x = u and y = v in inequality (3), then we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

then we have

M(Au, Bv, [phi](t)) [greater than or equal to] M(Au, Bv, t).

On the other hand, since M is non-decreasing, we get M(Au, Bv, [phi](t)) [less than or equal to] M(Au, Bv, t). Hence, M(Au, Bv, t) C for all t > 0. From Lemma 2.7, we conclude that C = 1, that is Au Bv. Moreover, if there is another point z such that Az = [P.sub.1][P.sub.3] ... [P.sub.2n-1]z. Then using inequality (3) it follows that Az [P.sub.1][P.sub.3] ... [P.sub.2n-1]z Bv = [P.sub.2][P.sub.4] ... [P.sub.2nv], or Au = Az. Hence, w = Au [P.sub.1][P.sub.3] ... [P.sub.2n-1]u is the unique point of coincidence of A and [P.sub.1][P.sub.3] ... [P.sub.2n-1]. From Lemma 2.11, it follows that w is the unique common fixed point of A and [P.sub.1][P.sub.3] ... [P.sub.2n-1]. By symmetry, q = Bv = [P.sub.2][P.sub.4] ... [P.sub.2nv] is the unique common fixed point of B and [P.sub.2][P.sub.4] ... [P.sub.2n]. Since w = q, we obtain that w is the unique common fixed point of B and [P.sub.2][P.sub.4] ... [P.sub.2n]. Now, we show that w is the fixed point of all the component mappings. Putting x = [P.sub.3] ... [P.sub.2n-1]w, y = w,[P'.sub.1] = [P.sub.1][P.sub.3] ... [P.sub.2n-1] and [P'.sub.2] = [P.sub.2][P.sub.4] ... [P.sub.2n] in inequality (3), we have

[ILLUSTRATION OMITTED]

thus, it follows that

M([P.sub.3] ... [P.sub.2n-1] w, w, [phi](t)) [greater than or equal to] M([P.sub.3] ... [P.sub.2n-1] w, w, t).

Since M is non-decreasing, we get M([P.sub.3] ... [P.sub.2n-1] w, w, [phi](t)) [less than or equal to] M([P.sub.3] ... [P.sub.2n-1] w, w, t). Hence, M([P.sub.3] ... [P.sub.2n-1] w, w, t) = C for all t > 0. From Lemma 2.7 we conclude that C = 1, that is [P.sub.3] ... [P.sub.2n-1] w = w. Hence, [P.sub.1] w = w. Continuing this procedure, we have

Aw = [P.sub.1]w = [P.sub.3]w = ... = [P.sub.2n-1] w = w.

So, Bw = [P.sub.2]w = [P.sub.4]w = ... = [P.sub.2n]w = w. So, w is the unique common fixed point of [P.sub.1], [P.sub.2], ..., [P.sub.2n], A and B.

The following result is a slight generalization of Theorem 3.4.

Corollary 3.5. Let [{[T.sub.[zeta]]}.sub.[zeta][member of]J] and [{[P.sub.i]}.sup.2n.sub.i=1] be two families of self maps on fuzzy metric space (X, M, *) where * is a continuous t-norm with a * a [greater than or equal to] a for all a [member of] [0, 1]. Further, let the pairs ([T.sub.[zeta]], [P.sub.1][P.sub.3] ... [P.sub.2n-1]) and ([T.sub.[zeta]], [P.sub.2][P.sub.4] ... [P.sub.2n]) are each owc satisfying: for a fixed [xi] [member of] J,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

for all x, y [member of] X and t > 0. Here, the function [phi](t) : [0, [infinity]) [right arrow] [0, [infinity]) is onto, strictly increasing and satisfies condition ([PHI]). Suppose that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

similarly,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Then all {[P.sub.i]} and {[T.sub.[zeta]]} have a unique common fixed point in X.

Remark 3.6. The conclusions of our results remain true if we take [phi](t) = kt, where k [member of] (0,1).

Acknowledgment

The authors would like to express their sincere thanks to Professor B. E. Rhoades [28,29] for his papers.

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Sunny Chauhan ([dagger]) and Suneel Kumar ([double dagger])

([dagger]) R. H. Government Postgraduate College, Kashipur, U. S. Nagar, 244713, Uttarakhand, India

([double dagger]) Government Higher Secondary School, Sanyasiowala PO-Jaspur, U. S. Nagar, 244712, Uttarakhand, India

E-mail: sun.gkv@gmail.com ksuneeLmath@rediffmail.com
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