Coupled fixed point theorems in cone metric spaces with a c-distance and applications

Erdal Karapinar,Wutiphol Sintunavarat,Poom Kumam

doi:10.1186/1687-1812-2012-194

Erdal Karapinar, Wutiphol Sintunavarat + Show 1 more

Open Access

https://doi.org/10.1186/1687-1812-2012-194

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Abstract

Abstract In this paper, we extend the very recent result of Sintunavarat et al. in the paper ‘Coupled fixed point theorems for weak contraction mapping under F-invariant set’ (Abstr. Appl. Anal. 2012:324874, 2012). In particular, we give an example of a nonlinear contraction mapping for which our result successfully detects a coupled fixed point in contrast to the result of Sintunavarat et al., which is not applied to show the existence of a coupled fixed point. As a consequence, the main results in this paper extend and unify many results in the topic of coupled fixed points including the results of Sintunavarat et al. Also, some applications of the main results are given. MSC: 54H25, 47H10.

Highlights

Let X be an arbitrary nonempty set and f : X → X be a mapping
Fixed point theorems are vital for the existence and uniqueness of differential equations, matrix equations, and integral equations
The function f : (, ] × [, ∞) × [, ∞) → [, ∞) is continuous, limt→ + f (t, ·, ·) = ∞ for all t ∈. Because of their important roles in the study of the existence and uniqueness of solutions of the periodic boundary value problems, nonlinear integral equations and the existence and uniqueness of positive solutions for the singular nonlinear fractional differential equations with boundary value, discussions on coupled fixed point theorems are of interest to many scientists

Summary

Introduction

Let X be an arbitrary nonempty set and f : X → X be a mapping. A fixed point for f is a point x ∈ X such that fx = x. The function f : ( , ] × [ , ∞) × [ , ∞) → [ , ∞) is continuous, limt→ + f (t, ·, ·) = ∞ for all t ∈ Because of their important roles in the study of the existence and uniqueness of solutions of the periodic boundary value problems, nonlinear integral equations and the existence and uniqueness of positive solutions for the singular nonlinear fractional differential equations with boundary value, discussions on coupled fixed point theorems are of interest to many scientists. Cho et al [ ] established new coupled fixed point theorems under contraction mappings by using the concept of the mixed monotone property and c-distance in partially ordered cone metric spaces as follows.

This implies

This implies that