Fixed Point Theorem for Monotone Non-Expansive Mappings

doi:10.28924/2291-8639-19-2021-288

Abstract

In this paper, we study the fixed point theorem for monotone nonexpansive mappings in the setting of a uniformly smooth and uniformly convex smooth Banach space.

Highlights

Given a complete metric space (X, d), the most well-studied types of self-maps are referred to as Lipschitz mappings, which are given by the metric inequality (1.1)d(T x, T y) ≤ kd(x, y), for all x, y ∈ X, where k > 0 is a real number, usually referred to as the Lipschitz constant of T
Let T be a nonexpansive self-mapping on a compact star-convex subset of a Banach space
We say that the normalized duality map J of a Banach space X is sequentially weakly continuous if a sequence {xn}n≥1 in X is weakly convergent to x, the sequence {J xn}n≥1 in X ∗ is weak-star convergent to J x

Summary

Introduction

Given a complete metric space (X , d), the most well-studied types of self-maps are referred to as Lipschitz mappings (or Lipschitz maps, for short), which are given by the metric inequality (1.1)d(T x, T y) ≤ kd(x, y), for all x, y ∈ X , where k > 0 is a real number, usually referred to as the Lipschitz constant of T. Fixed points problems of contraction mappings always exist Monotone nonexpansive mappings; normalised duality mappings; uniformly convex spaces; uniformly smooth spaces; fixed points.

Results

Conclusion