Fixed point theorems for fuzzy mappings and applications to ordinary fuzzy differential equations

Hemant Kumar Nashine,Wiyada Kumam,Calogero Vetro,Poom Kumam

doi:10.1186/1687-1847-2014-232

Abstract

Abstract Ran and Reurings (Proc. Am. Math. Soc. 132(5):1435-1443, 2004) proved an analog of the Banach contraction principle in metric spaces endowed with a partial order and discussed some applications to matrix equations. The main novelty in the paper of Ran and Reurings involved combining the ideas in the contraction principle with those in the monotone iterative technique. Motivated by this, we present some common fixed point results for a pair of fuzzy mappings satisfying an almost generalized contractive condition in partially ordered complete metric spaces. Also we give some examples and an application to illustrate our results. MSC:46S40, 47H10, 34A70, 54E50.

Highlights

The Banach contraction principle [ ] is a very popular tool in solving existence problems in many branches of mathematical analysis
A new category of contractive fixed point problems was addressed by Khan et al [ ] that introduced the concept of altering distance function, which is a control function that alters distance between two points in a metric space
We prove a common fixed point theorem for a pair of fuzzy mappings without taking into account any commutativity condition in complete ordered metric spaces

Summary

Introduction

The Banach contraction principle [ ] is a very popular tool in solving existence problems in many branches of mathematical analysis. Existence theorems of fixed points have been established for mappings defined on various types of spaces and satisfying different types of contractive inequalities. Many results appeared related to fixed points in complete metric spaces endowed with a partial ordering. ]. In their paper, Ran and Reurings proved an analog of the Banach contraction principle in a metric space endowed with a partial ordering and gave applications to matrix equations. A number of papers appeared in which fixed points of fuzzy mappings satisfying contractive inequalities have been discussed (see [ – ] and references therein). We prove a common fixed point theorem for a pair of fuzzy mappings without taking into account any commutativity condition in complete ordered metric spaces. X, y ∈ X are called comparable if x y or y x holds

Main results

Conclusions