Coupled fixed points for multivalued mappings in fuzzy metric spaces
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RESEARCH
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Coupled fixed points for multivalued mappings in fuzzy metric spaces Zheyong Qiu and Shihuang Hong* *
Correspondence: [email protected] Institute of Applied Mathematics and Engineering Computations, Hangzhou Dianzi University, Hangzhou, 310018, People’s Republic of China
Abstract In this paper, we establish two coupled fixed point theorems for multivalued nonlinear contraction mappings in partially ordered fuzzy metric spaces. The theorems presented extend some corresponding results due to ordinary metric spaces. An example is given to illustrate the usability of our results. Keywords: coupled fixed point; multivalued contractive mapping; fuzzy metric space; partially ordered set
1 Introduction In , Nadler [] extended the famous Banach contraction principle from single-valued mappings to multivalued mappings and proved the existence of fixed points for contractive multivalued mappings in complete metric spaces. Since then, the existence of fixed points for various multivalued contractive mappings has been studied by many authors under different conditions. For details, we refer to [–] and the references therein. For instance, in [] Ćirić has proved a fixed point theorem for the single-valued mappings satisfying some contractive condition. Samet and Vetro [] extended this result to multivalued mappings and proved the existence of a coupled fixed point theorem for the multivalued contraction. One of the most important problems in fuzzy topology is to obtain an appropriate concept of fuzzy metric spaces. This problem has been investigated by many authors from different points of view. In particular, George and Veeramani [, ] introduced and studied the notion of fuzzy metric M on a set X with the help of continuous t-norms introduced in [], and from now on, when we talk about fuzzy metrics, we refer to this type. Fuzzy metric spaces have many applications. In particular, on the fuzzy metric space, by using some topological properties induced by this kind of fuzzy metrics, there are several fixed point results established. Some instances of these works are in [–]. In fact, fuzzy fixed point results are more versatile than the regular metric fixed point results. This is due to the flexibility which the fuzzy concept inherently possesses. For example, the Banach contraction mapping principle has been extended in fuzzy metric spaces in two inequivalent ways in [, ]. Fuzzy fixed point theory has a developed literature and can be regarded as a subject in its own right (see []). In recent times, the existence of common or coupled fixed points of a fuzzy version for multiple mappings has attracted much attention. We mention that the coupled fixed point results were proved by Sedghi et al. [], which is a fuzzy version of the result of []. Choudhury [] further extended the result of [] and provided the existence results of coupled coincidence points for compatible mappings in partially ordered fuzzy © 2013 Qiu and Hong; licensee Springer. This is an Open Access article distributed un
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