Abstract

We prove a fixed point theorem for mappings satisfying an implicit relation in a complete fuzzy metric space.

1. Introduction and Preliminaries

The concept of a fuzzy set was introduced by Zadeh [1] in 1965. This concept was used in topology and analysis by many authors. George and Veeramani [2] modified the concept of fuzzy metric space introduced by Kramosil and Michalek [3] and defined the Hausdorff topology of fuzzy metric spaces, which have important applications [4] in quantum particle physics. Afterwards, Beg and Abbas [5], Vasuki [6], Popa [7], and Imad et al. [8, 9] obtained results about fixed point and invariant approximation in fuzzy metric spaces. The aim of this paper is to obtain fixed point of mapping satisfying an implicit relation on fuzzy metric spaces. Our results generalize several known results in the literature.

First, we give some basic definitions.

Definition 1 (see [10]). A binary operation is called a continuous -norm if is an abelian topological monoid, that is,(1) is associative and commutative,(2) is continuous,(3) for all ,(4) whenever and , for each .

Four typical examples of a continuous -norms are , for and , .

Definition 2 (see [2]). The 3-tuple is called a fuzzy metric space if is an arbitrary nonempty set, is a continuous -norm, and is a fuzzy set on satisfying the following conditions, for each and ,  (FM-1) ,  (FM-2) if and only if ,  (FM-3) ,  (FM-4) ,  (FM-5) is continuous.

Let be a fuzzy metric space. For , the open ball with a center and a radius is defined by

A subset is called open if, for each , there exist and such that . Let denote the family of all open subsets of . Then, is called the topology on induced by the fuzzy metric . This topology is Hausdorff and first countable. A subset of is said to be -bounded if there exist and such that for all .

Example 3. Let . Denote for all . For each , define for all . Then, is a fuzzy metric space.

Example 4. Let be an ordinary metric space and be an increasing and continuous function from into such that . Four typical examples of these functions are , , , and . Let for all . For each , define for all . It is easy to see that is a fuzzy metric space.

Definition 5 (see [2]). Let be a fuzzy metric space.(i) A sequence in is said to be convergent to a point if for all .(ii) A sequence in is called Cauchy sequence if, for each and , there exits such that for each .(iii) A fuzzy metric space in which every Cauchy sequence is convergent is said to be complete.(iv) A fuzzy metric space in which every sequence has a convergent subsequence is said to be compact.

Lemma 6 (see [11]). For all , is a nondecreasing function.

Definition 7. Let be a fuzzy metric space. is said to be continuous on if whenever is a sequence in which converges to a point , that is,

Lemma 8 (see [11]). is a continuous function on .

2. Main Results

Let be a fuzzy metric space and . Define for all . For an in fuzzy metric space , let . Then, is finite for all , is nonincreasing, for some , and also for all .

Let be the set of all continuous functions such that is nondecreasing on satisfying the following conditions.

implies that where is a nondecreasing continuous function with for .

Example 9. , defined the following:(i),(ii) such that for every , , where for .
In this paper, our main result is the following theorem.

Theorem 10. Let be a complete bounded fuzzy metric space and a self map of satisfying for all the implicit relation(i) where .
Then, has a unique fixed point in , and is continuous at .

Proof. Let and . Let , where . Then, we know for some . If for some , then has a fixed point, . Assume that for each . Let be fixed. Taking , in (i) where and , we have
Thus, we have since is nondecreasing on . Also, since is nonincreasing, we have which implies that
Thus, for all , we have . Letting , we get . If , then , which is a contradiction. Thus and hence . Thus, given , there exists an such that . Then, we have for and , . Therefore, is a Cauchy sequence in . By the completeness of , there exists a such that . Taking in (i), we have
Taking , we have which implies that . Hence, . For the uniqueness, let and be fixed points of  . Taking , in (i), we have
Since is nondecreasing on , we have which implies that , which is a contradiction. Thus, we have . Now, we show that is continuous at . Let be a sequence in and . Taking in (i), we have which implies that . Taking limit inf, we have
Thus, . Hence, is continuous at .

Corollary 11. Let be a complete bounded fuzzy metric space, , and a self map of satisfying for all ,(i) where .
Then, has a unique fixed point in , and is continuous at .

Proof. From Theorem 10, has a unique fixed point in and is continuous at . Since , is also a fixed point of . By the uniqueness, it follows .

In Theorem 10 (Corollary 11, resp.), If we take , then we have the next result.

Corollary 12. Let be a complete bounded fuzzy metric space and a self map of satisfying for all ,(i)
Then, has a unique fixed point in , and is continuous at .

Example 13. Let . Define by for all and . Then, is a complete fuzzy metric space where . Define map by for and let defined by . It is easy to see that
Thus, satisfy all the hypotheses of Corollary 12, and hence has a unique fixed point. Indeed, is the unique fixed point of .

Corollary 14. Let be a complete bounded fuzzy metric space, , and a self map of satisfying for all ,(i)
Then, has a unique fixed point in , and is continuous at .

Corollary 15. Let be a complete bounded fuzzy metric space and a self map of satisfying for all ,(i) such that for every , . Then, has a unique fixed point in , and is continuous at .
Let be the class of all continuous functions such that is nondecreasing on satisfying implies that .

Example 16. Let defined by for some . Then, is continuous on and nondecreasing on . Also, we have that implies that . Thus .

Theorem 17. Let be a compact fuzzy metric space and a continuous self map of satisfying for all with(i) where .
Then, has a unique fixed point in .

Proof. We know that for is compact and . Let . Then, is a nonempty compact subset of , and . We claim that is a singleton set. Suppose is not singleton. Then, we know that the function has a minimum value. That is, there exists a such that for all . Since , there exist such that . Thus, we have and so which implies that . This is a contradiction. Thus, is singleton, and hence has a fixed point in . Uniqueness of fixed point of follows from (i).

Corollary 18. Let be a compact fuzzy metric space, , and a continuous self map of satisfying for all ,(i) where .
Then, has a unique fixed point in .

Proof. From Theorem 17, has a unique fixed point in . Since , is also a fixed point of . By the uniqueness, it follows .