On locally contractive fuzzy set-valued mappings
© Ahmad et al.; licensee Springer. 2014
Received: 28 October 2013
Accepted: 20 January 2014
Published: 13 February 2014
We prove the existence of common fuzzy fixed points for a sequence of locally contractive fuzzy mappings satisfying generalized Banach type contraction conditions in a complete metric space by using iterations. Our main result generalizes and unifies several well-known fixed-point theorems for multivalued maps. Illustrative examples are also given.
MSC:46S40, 47H10, 54H25.
The Banach contraction theorem and its subsequent generalizations play a fundamental role in the field of fixed point theory. In particular, Heilpern introduced in  the notion of a fuzzy mapping in a metric linear space and proved a Banach type contraction theorem in this framework. Subsequently several other authors [2–10] have studied and established the existence of fixed points of fuzzy mappings. The aim of this paper is to prove a common fixed-point theorem for a sequence of fuzzy mappings in the context of metric spaces without the assumption of linearity. Our results generalize and unify several typical theorems of the literature.
for all .
According to Heilpern , a fuzzy set A in a metric linear space is said to be an approximate quantity if is compact and convex in X, for each , and . The family of all approximate quantities of the metric linear space is denoted by .
It is well known that is a metric on .
whenever and , .
In the sequel the letter ℕ will denote the set of positive integer numbers.
The following well-known properties on the Hausdorff metric (see e.g. ) will be useful in the next section.
Lemma 2.1 Let be a metric space and let with , . If , then there exists such that .
Lemma 2.2 Let be a metric space and let be a sequence in such that , for some . If , for all , and , then .
Now, let X be an arbitrary set and let Y be a metric space. A mapping T is called fuzzy mapping if T is a mapping from X into . In fact, a fuzzy mapping T is a fuzzy subset on with membership function . The value is the grade of membership of ξ in .
If is a metric space and T is a (fuzzy) mapping from X into , we say that is a fixed point of T if .
We conclude this section with the notion of contractiveness that will be used in our main result.
Definition 2.3 (compare )
Let . A function is said to be a MT-function if it satisfies Mizoguchi-Takahashi’s condition (i.e., , for all ).
Clearly, if is a nondecreasing function or a nonincreasing function, then it is a MT-function. So the set of MT-functions is a rich class.
3 Fixed points of fuzzy mappings
Fixed-point theorems for locally contractive mappings were studied, among others, by Edelstein , Beg and Azam , Holmes , Hu , Hu and Rosen , Ko and Tasi , Kuhfitting  and Nadler .
Heilpern  established a fixed-point theorem for fuzzy contraction mappings in metric linear spaces, which is a fuzzy extension of Banach’s contraction principle. Afterwards Azam et al. [4, 5], and Lee and Cho  further extended Banach’s contraction principle to fuzzy contractive mappings in Heilpern’s sense. In our main result (Theorem 3.1 below) we establish a common fixed-point theorem for a sequence of generalized fuzzy uniformly locally contraction mappings on a complete metric space without the requirement of linearity. This is a generalization of many conventional results of the literature.
Let , and . A metric space is said to be ε-chainable if given , there exists an ε-chain from ζ to ξ (i.e., a finite set of points , such that , for all ). A mapping is called an uniformly locally contractive mapping if and , implies . A mapping is called an uniformly locally contractive fuzzy mapping if and , imply . We remark that a globally contractive mapping can be regarded as an uniformly locally contractive mapping and for some special spaces every locally contractive mapping is globally contractive.
for all , where is a MT-function, then the sequence has a common fixed point, i.e., there is such that , for all .
be an arbitrary ε-chain from to . (We suppose, without loss of generality, that , for each with .)
for all .
By assumption, , so there exists such that , for all where .
Since , for all , it follows that is a Cauchy sequence. Since is complete, there is such that . So for each there is such that implies . This in view of inequality (1) implies , for all . Consequently, . Since with , we deduce from Lemma 2.2 that , for all . This completes the proof. □
for all , where , then the sequence has a common fixed point.
Proof Apply Theorem 3.1 when ψ is the MT-function defined as , for all . □
for all , where is a MT-function. Then the sequence has a common fixed point.
Proof Since and , for all , the result follows immediately from Theorem 3.1. □
for all , where . Then the sequence has a common fixed point.
Corollary 3.5 
for , where is a MT-function. Then and have a common fixed point.
Let , a complete ε-chainable metric linear space and T: an uniformly locally contractive fuzzy mapping. Then T has a fixed point.
where is a MT-function. Then S has a fixed point.
for all , we deduce that condition (1) of Theorem 3.1 is satisfied for T. Hence T has a fixed point , i.e., . We conclude that . The proof is complete. □
Corollary 3.8 
where . Then S has a fixed point.
for all . Then S has a fixed point in X.
Proof Apply Corollary 3.8 with . □
We conclude the paper with two examples to support Theorem 3.1 and Corollary 3.2.
Hence, for , the conditions of Corollary 3.2, and hence of Theorem 3.1, are satisfied for any , whereas X is not linear. Therefore all previous relevant fixed point results Corollaries 3.3-3.6 on metric linear spaces are not applicable.
Example 3.11 Let be the complete metric space such that , , for all , and whenever (in the sequel we shall write instead of ).
Note that a sequence is a Cauchy sequence in if and only if . Moreover, is the only non-isolated point of X for the topology induced by d.
Therefore , for all and (recall that each is an isolated point for the induced topology, so every bounded interval belongs to ).
We show that condition (1) of Theorem 3.1 is satisfied for and ψ as defined above. Indeed, let with and . Assume without loss of generality that .
We have shown that all conditions of Theorem 3.1 are satisfied (in fact is the only fixed point of T).
The third author thanks the support of the Ministry of Economy and Competitiveness of Spain, Grant MTM2012-37894-C02-01.
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