Tripled fixed point theorem in fuzzy metric spaces and applications

A Roldán,J Martínez-Moreno,C Roldán

doi:10.1186/1687-1812-2013-29

A Roldán, J Martínez-Moreno + Show 1 more

Open Access

https://doi.org/10.1186/1687-1812-2013-29

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Abstract

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.

Highlights

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 established some interesting coupled fixed point theorems
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
Fixed point theorems have been studied in many contexts, one of which is the fuzzy setting

Summary

Introduction

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 established some interesting coupled fixed point theorems. 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. There exists considerable literature about fixed point properties for mappings defined on fuzzy metric spaces, which have been studied by many authors (see [ , – ]).

Results

Conclusion