Abstract
Recently Samet, Vetro and Vetro introduced the notion of α-ψ-contractive type mappings and established some fixed point theorems in complete metric spaces. In this paper, we introduce the notion of -contractive multifunctions and give a fixed point result for these multifunctions. We also obtain a fixed point result for self-maps in complete metric spaces satisfying a contractive condition.
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1 Introduction
Fixed point theory has many applications in different branches of science. During the last few decades, there has been a lot of activity in this area and several well-known fixed point theorems have been extended by a number of authors in different directions (see, for example, [1–38]). Recently Samet, Vetro and Vetro introduced the notion of α-ψ-contractive type mappings [33]. Denote with Ψ the family of nondecreasing functions such that for all , where is the n th iterate of ψ. It is known that for all and [33]. Let be a metric space, T be a self-map on X, and be a function. Then T is called an α-ψ-contraction mapping whenever for all . Also, we say that T is α-admissible whenever implies [33]. Also, we say that α has the property (B) if is a sequence in X such that for all and , then for all . Let be a complete metric space and T be an α-admissible α-ψ-contractive mapping on X. Suppose that there exists such that . If T is continuous or T has the property (B), then T has a fixed point (see [33]; Theorems 2.1 and 2.2). Finally, we say that X has the property (H) whenever for each there exists such that and . If X has the property (H) in Theorems 2.1 and 2.2, then X has a unique fixed point ([33]; Theorem 2.3). It is considerable that the results of Samet et al. generalize similar ordered results in the literature (see the results of the third section in [33]). The aim of this paper is to introduce the notion of -contractive multifunctions and give a fixed point result about the multifunctions. Let be a metric space, be a closed-valued multifunction, and be a function. In this case, we say that T is an -contractive multifunction whenever for , where H is the Hausdorff generalized metric, and denotes the family of all nonempty subsets of X. Also, we say that T is -admissible whenever implies .
Example 1.1 Let , and be a fixed number. Define by for all and by whenever and whenever or . Now, we show that T is -admissible. If , then and so Tx and Ty are subsets of . Thus, for all and . Hence, for all and . This implies that
Therefore, T is -admissible. Now, we show that T is an -contractive multifunction, where for all . If or , then an easy calculation shows us that . If , then . By using the definition of the Hausdorff metric, it is easy to see that for . Thus, for . Therefore, T is an -contractive multifunction.
Let be an ordered metric space and . We say that whenever for each there exists such that . Also, we say that whenever for each and we have .
2 Main results
Now, we are ready to state and prove our main results. In the following result, we use the argument similar to that in the proof of Theorem 3.1 in [22].
Theorem 2.1 Letbe a complete metric space, be a function, be a strictly increasing map and T be a closed-valued, -admissible and-contractive multifunction on X. Suppose that there existandsuch that. Assume that ifis a sequence in X such thatfor all n and, thenfor all n. Then T has a fixed point.
Proof If , then we have nothing to prove. Let . If , then is a fixed point of T. Let and be given. Then
Hence, there exists such that
It is clear that and . Thus, . Now, put . Then, and . Since ψ is strictly increasing, . Put . Then . If , then is a fixed point of T. Assume that . Then
Hence, there exists such that
It is clear that , and . Now, put . Then . If , then is a fixed point of T. Assume that . Then
Thus, there exists such that
By continuing this process, we obtain a sequence in X such that , , and for all n. Now, for each , we have
Hence, is a Cauchy sequence in X. Choose such that . Since for all n and T is -admissible, for all n, thus
for all n. Therefore, and so . □
Example 2.1 Let and . Define by for all , for all and by whenever and whenever or . Then it is easy to check that T is an -admissible and -contractive multifunction, where for all . Put and . Then . Also, if is a sequence in X such that for all n and , then for all n. Note that T has infinitely many fixed points.
Corollary 2.2 Letbe a complete ordered metric space, be a strictly increasing map and T be a closed-valued multifunction on X such that
for allwith. Suppose that there existsandsuch that. Assume that ifis a sequence in X such thatfor all n and, thenfor all n. Ifimplies, then T has a fixed point.
Proof Define by whenever and whenever . Since implies , implies . Thus, it is easy to check that T is an -admissible and -contractive multifunction on X. Now, by using Theorem 2.1, T has a fixed point. □
Now, we prove the following result for self-maps.
Theorem 2.3 Letbe a complete metric space, be a function, and T be a self-map on X such thatfor all, where. Suppose that T is α-admissible and there existssuch that. Assume that ifis a sequence in X such thatfor all n and, thenfor all n. Then T has a fixed point.
Proof Take such that and define the sequence in X by for all . If for some n, then is a fixed point of T. Assume that for all n. Since T is α-admissible, it is easy to check that for all natural numbers n. Thus, for each natural number n, we have
If , then
which is contradiction. Thus, for all n. Hence, and so for all n. It is easy to check that is a Cauchy sequence. Thus, there exists such that . By using the assumption, we have for all n. Thus,
for sufficiently large n. Hence, and so . □
Example 2.2 Let and . Define the self-map T on X by for , for and by whenever and whenever or . Then it is easy to check that T is α-admissible and for all , where for all . Also, and if is a sequence in X such that for all n and , then for all n. Note that, T has two fixed points.
Corollary 2.4 Letbe a complete ordered metric space, and T be a self-map on X such thatfor allwith. Suppose that there existssuch that. Ifis a sequence in X such thatfor all n and, thenfor all n. Ifimplies, then T has a fixed point.
If we substitute a partial metric ρ for the metric d in Theorem 2.3, it is easy to check that a similar result holds for the partial metric case as follows.
Theorem 2.5 Letbe a complete partial metric space, be a function, and T be a self-map on X such thatfor all, where. Suppose that T is α-admissible and there existssuch that. Assume that ifis a sequence in X such thatfor all n and, thenfor all n. Then T has a fixed point.
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Asl, J.H., Rezapour, S. & Shahzad, N. On fixed points of α-ψ-contractive multifunctions. Fixed Point Theory Appl 2012, 212 (2012). https://doi.org/10.1186/1687-1812-2012-212
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DOI: https://doi.org/10.1186/1687-1812-2012-212