Contraction mapping principle with generalized altering distance function in ordered metric spaces and applications to ordinary differential equations

Yongfu Su

doi:10.1186/1687-1812-2014-227

Abstract

The aim of this paper is to present the definition of a generalized altering distance function and to extend the results of Yan et al. (Fixed Point Theory Appl. 2012:152, 2012) and some others, and to prove a new fixed point theorem of generalized contraction mappings in a complete metric space endowed with a partial order by using generalized altering distance functions. The results of this paper can be used to investigate a large class of nonlinear problems. As an application, we discuss the existence of a solution for a periodic boundary value problem.

Highlights

The Banach contraction mapping principle is a classical and powerful tool in nonlinear analysis
Weak contractions are generalizations of the Banach contraction mapping, which have been studied by several authors
In [ – ], the authors prove some types of weak contractions in complete metric spaces, respectively

Summary

Introduction

The Banach contraction mapping principle is a classical and powerful tool in nonlinear analysis. Harjani and Sadarangani proved some fixed point theorems for weak contractions and generalized contractions in partially ordered metric spaces by using the altering distance function in [ , ], respectively. Let T : X → X be a non-decreasing mapping such that ψ d(Tx, Ty) ≤ φ d(x, y) , ∀x ≥ y, where ψ is a generalized altering distance functions and φ: [ , ∞) → [ , ∞) is a right upper semi-continuous function with the condition ψ(t) > φ(t) for all t >. Ψ is a generalized altering distance function and we have the condition ψ(t) > φ(t) for t > , this gives us that {d(z, f n(x))} is a non-negative decreasing sequence and, there exists γ such that lim d z, Tn(x) = γ.

This problem is equivalent to the integral equation

The first integral gives us t

Proof Consider the cone

It is easy to verify that