Vector-valued metrics, fixed points and coupled fixed points for nonlinear operators

Adrian Petruşel,Gabriela Petruşel,Cristina Urs

doi:10.1186/1687-1812-2013-218

Adrian Petruşel, Gabriela Petruşel + Show 1 more

Open Access

https://doi.org/10.1186/1687-1812-2013-218

Copy DOI

Abstract

Abstract In this paper, we will present fixed point theorems for singlevalued and multivalued operators in spaces endowed with vector-valued metrics, as well as a Gnana Bhaskar-Lakshmikantham-type theorem for the coupled fixed point problem, associated to a pair of singlevalued operators (satisfying a generalized mixed monotone property) in ordered metric spaces. The approach is based on Perov-type fixed point theorems in spaces endowed with vector-valued metrics. The Ulam-Hyers stability and the limit shadowing property of the fixed point problems are also discussed. MSC:47H10, 54H25.

Highlights

The classical Banach contraction principle is a very useful tool in nonlinear analysis with many applications to operatorial equations, fractal theory, optimization theory and other topics
Banach contraction principle was extended for singlevalued contraction on spaces endowed with vector-valued metrics by Perov [ ] and Perov and Kibenko [ ]
The case of multivalued contractions on spaces endowed with vector-valued metrics is treated in [ – ], etc

Summary

Introduction

The classical Banach contraction principle is a very useful tool in nonlinear analysis with many applications to operatorial equations, fractal theory, optimization theory and other topics. (Perov) Let (X, d) be a complete generalized metric space, and let the operator f : X → X be with the property that there exists a matrix A ∈ Mmm(R+) convergent towards zero such that d f (x), f (y) ≤ Ad(x, y), for all x, y ∈ X. Our second purpose is to present, in the setting of an ordered metric space, a Gnana BhaskarLakshmikantham-type theorem for the coupled fixed point problem associated to a pair of singlevalued operators satisfying a generalized mixed monotone assumption. (see [ , ]) If (X, d) is a generalized metric space, f : X → X is called a weakly Picard operator if and only if the sequence (f n(x))n∈N of successive approximations of f converges for all x ∈ X and the limit (which may depend on x) is a fixed point of f. Notice that d is a generalized metric on X if and only if di are metrics on X for each i ∈ { , , . . . , m}

We denote by

If we define