Fixed points of monotone mappings and application to integral equations

Abstract

In this work, we discuss the existence of fixed points of monotone nonexpansive mappings defined on partially ordered Banach spaces. This work is a continuity of the previous works of Ran and Reurings, Nieto et al., and Jachimsky done for contraction mappings. As an application, we discuss the existence of solutions to an integral equations.MSC: Primary 46B20; 45D05; secondary 47E10; 34A12

The Knaster–Tarski theorem versus monotone nonexpansive mappings

“…As it is shown in [2, Section 3], we can associate to this integral equation the operator J : L 2 (Ω, ) → L 2 (Ω, ) given by (Jy)(t) = g(t) + ΩF (t)(y)(s) dµ(s) whereF (t)(y)(s) = F (t, s, y(s)). Then it can be shown, in the same terms as in [2], that J sends the whole L 2 (Ω, ) to a closed ball of sufficiently large radius. Now, if we consider the weak topology as τ in Theorem 2 it can be shown that order intervals are closed with respect to this topology.…”

A fixed point theorem for monotone asymptotically nonexpansive mappings

“…In all these works, the mappings considered are monotone contractions. The case of monotone nonexpansive mappings was first considered in [1]. Then the race was on to find out whether the classical fixed point theorems for nonexpansive mappings still hold for monotone nonexpansive mappings.…”

On monotone mappings in modular function spaces

“…In the next example, we illustrate these ideas and show how to apply our Theorem 3.1. This example was inspired from the one used in [2]. The fundamental difference resides in the fact that most uniformly convex spaces, like L p , fail to satisfy the Opial property as a key assumption in the paper [2].…”