Fixed point variational solutions for uniformly continuous pseudocontractions in Banach spaces

Aniefiok Udomene

doi:10.1155/fpta/2006/69758

Abstract

Let be a reflexive Banach space with a uniformly Gâteaux differentiable norm, let be a nonempty closed convex subset of , and let be a uniformly continuous pseudocontraction. If is any contraction map on and if every nonempty closed convex and bounded subset of has the fixed point property for nonexpansive self-mappings, then it is shown, under appropriate conditions on the sequences of real numbers , , that the iteration process , , , strongly converges to the fixed point of , which is the unique solution of some variational inequality, provided that is bounded.

Highlights

Let E be a real Banach space with dual E∗ and K a nonempty closed convex subset of E
The mapping T is called nonexpansive if L = 1 and is called a contraction if L < 1
The example, T(x) = 1 − x2/3, 0 ≤ x ≤ 1, is a continuous pseudocontraction which is not nonexpansive. It follows from a result of Kato [3] that T is pseudocontractive if and only if there exists j(x − y) ∈ J(x − y) such that Tx − T y, j(x − y) ≤ x − y 2, ∀x, y ∈ D(T)

Summary

Introduction

Let E be a real Banach space with dual E∗ and K a nonempty closed convex subset of E. Following Morales [6], a mapping T with domain D(T) and range ᏾(T) in E is called strongly pseudocontractive if for some constant k < 1 and ∀x, y ∈ D(T),. The example, T(x) = 1 − x2/3, 0 ≤ x ≤ 1, is a continuous pseudocontraction which is not nonexpansive. It follows from a result of Kato [3] that T is pseudocontractive if and only if there exists j(x − y) ∈ J(x − y) such that Tx − T y, j(x − y) ≤ x − y 2, ∀x, y ∈ D(T).

Uniformly continuous pseudocontractions

Preliminaries

Main results