Cyclic (noncyclic) phi-condensing operator and its application to a system of differential equations

Moosa Moosa Gabeleh,Calogero Vetro,Philemon Moshokoa

doi:10.15388/na.2019.6.8

Abstract

We establish a best proximity pair theorem for noncyclic φ-condensing operators in strictly convex Banach spaces by using a measure of noncompactness. We also obtain a counterpart result for cyclic φ-condensing operators in Banach spaces to guarantee the existence of best proximity points, and so, an extension of Darbo’s fixed point theorem will be concluded. As an application of our results, we study the existence of a global optimal solution for a system of ordinary differential equations.

Highlights

Let X be a Banach space and C ⊆ X
In 2005, the following existence theorem of best proximity pairs for noncyclic relatively nonexpansive mappings was established
We introduce new classes of noncyclic mappings, called noncyclic φ-condensing operators, and investigate the existence of best proximity pairs by using a notion of measure of noncompactness

Summary

Introduction

Let X be a Banach space and C ⊆ X. It is well known that if C is a nonempty, compact and convex subset of a Banach space X, any nonexpansive mapping of C into C has a fixed point. In 2005, the following existence theorem of best proximity pairs for noncyclic relatively nonexpansive mappings was established. We introduce new classes of noncyclic (cyclic) mappings, called noncyclic (cyclic) φ-condensing operators, and investigate the existence of best proximity pairs (points) by using a notion of measure of noncompactness. In this way, we show that the results alike to the celebrated Darbo’s fixed point theorem for condensing mappings can be obtained for cyclic φ-condensing operators. As an application of our main conclusions, we prove the existence of a global optimal solution for a system of differential equations

Preliminaries

Noncyclic φ-condensing operators

Existence of optimal solution for a system of differential equations