Tripled fixed point theorem in fuzzy metric spaces and applications
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Tripled fixed point theorem in fuzzy metric spaces and applications A Roldán1 , J Martínez-Moreno2* and C Roldán3 *
Correspondence: [email protected] 2 Department of Mathematics, University of Jaén, Jaén, Spain Full list of author information is available at the end of the article
Abstract In this paper we prove an existence and uniqueness theorem for contractive type mappings in fuzzy metric spaces. In order to do that, we consider a slight modification of the concept of a tripled fixed point introduced by Berinde et al. (Nonlinear Anal. TMA 74:4889-4897, 2011) for nonlinear mappings. Additionally, we obtain some fixed point theorems for metric spaces. These results generalize, extend and unify several classical and very recent related results in literature. For instance, we obtain an extension of Theorem 4.1 in (Zhu and Xiao in Nonlinear Anal. TMA 74:5475-5479, 2011) and a version in non-partially ordered sets of Theorem 2.2 in (Bhaskar and Lakshmikantham in Nonlinear Anal. TMA 65:1379-1393, 2006). As application, we solve a kind of Lipschitzian systems in three variables and an integral system. Finally, examples to support our results are also given.
Introduction In a recent paper, Bhaskar and Lakshmikantham [] introduced the concepts of coupled fixed point and mixed monotone property for contractive operators of the form F : X × X → X, where X is a partially ordered metric space, and then established some interesting coupled fixed point theorems. They also illustrated these important results by proving the existence and uniqueness of the solution for a periodic boundary value problem. Later, Lakshmikantham and Ćirić [] proved coupled coincidence and coupled common fixed point results for nonlinear mappings satisfying certain contractive conditions in partially ordered complete metric spaces. After that many results appeared on coupled fixed point theory (see, e.g., [–]). Fixed point theorems have been studied in many contexts, one of which is the fuzzy setting. The concept of fuzzy sets was initially introduced by Zadeh [] in . To use this concept in topology and analysis, many authors have extensively developed the theory of fuzzy sets and its applications. One of the most interesting research topics in fuzzy topology is to find an appropriate definition of fuzzy metric space for its possible applications in several areas. It is well known that a fuzzy metric space is an important generalization of the metric space. Many authors have considered this problem and have introduced it in different ways. For instance, George and Veeramani [] modified the concept of a fuzzy metric space introduced by Kramosil and Michalek [] and defined the Hausdorff topology of a fuzzy metric space. There exists considerable literature about fixed point properties for mappings defined on fuzzy metric spaces, which have been studied by many authors (see [, –]). Zhu and Xiao [] and Hu [] gave a coupled fixed point theorem for contractions in fuzzy metric spaces, and Fang [] proved some common fixed © 2013 Ro
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