Approximate fixed point sequences and convergence theorems for Lipschitz pseudocontractive maps

C. E. Chidume, H. Zegeye

Research output: Contribution to journalArticle

64 Citations (Scopus)

Abstract

Let K be a nonempty closed convex subset of a real Banach space E and T be a Lipschitz pseudocontractive self-map of K with F(T) := {x ε K : T x = x} ≠ Ø. An iterative sequence {xn} is constructed for which ∥xn - Txn∥ → 0 as n → ∞. If, in addition, AT is assumed to be bounded, this conclusion still holds without the requirement that F(T] ≠ Ø. Moreover, if, in addition, E has a uniformly Gâteaux differentiable norm and is such that every closed bounded convex subset of K has the fixed point property for nonexpansive self-mappings, then the sequence {xn} converges strongly to a fixed point of T, Our iteration method is of independent interest.

Original languageEnglish
Pages (from-to)831-840
Number of pages10
JournalProceedings of the American Mathematical Society
Volume132
Issue number3
Publication statusPublished - Mar 1 2004

Fingerprint

Approximate Fixed Point
Banach spaces
Set theory
Convergence Theorem
Lipschitz
Closed
Fixed Point Property
Subset
Iteration Method
Differentiable
Fixed point
Banach space
Converge
Norm
Requirements

All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Applied Mathematics

Cite this

@article{889ac697a3194919adc4d6c209421c39,
title = "Approximate fixed point sequences and convergence theorems for Lipschitz pseudocontractive maps",
abstract = "Let K be a nonempty closed convex subset of a real Banach space E and T be a Lipschitz pseudocontractive self-map of K with F(T) := {x ε K : T x = x} ≠ {\O}. An iterative sequence {xn} is constructed for which ∥xn - Txn∥ → 0 as n → ∞. If, in addition, AT is assumed to be bounded, this conclusion still holds without the requirement that F(T] ≠ {\O}. Moreover, if, in addition, E has a uniformly G{\^a}teaux differentiable norm and is such that every closed bounded convex subset of K has the fixed point property for nonexpansive self-mappings, then the sequence {xn} converges strongly to a fixed point of T, Our iteration method is of independent interest.",
author = "Chidume, {C. E.} and H. Zegeye",
year = "2004",
month = "3",
day = "1",
language = "English",
volume = "132",
pages = "831--840",
journal = "Proceedings of the American Mathematical Society",
issn = "0002-9939",
publisher = "American Mathematical Society",
number = "3",

}

Approximate fixed point sequences and convergence theorems for Lipschitz pseudocontractive maps. / Chidume, C. E.; Zegeye, H.

In: Proceedings of the American Mathematical Society, Vol. 132, No. 3, 01.03.2004, p. 831-840.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Approximate fixed point sequences and convergence theorems for Lipschitz pseudocontractive maps

AU - Chidume, C. E.

AU - Zegeye, H.

PY - 2004/3/1

Y1 - 2004/3/1

N2 - Let K be a nonempty closed convex subset of a real Banach space E and T be a Lipschitz pseudocontractive self-map of K with F(T) := {x ε K : T x = x} ≠ Ø. An iterative sequence {xn} is constructed for which ∥xn - Txn∥ → 0 as n → ∞. If, in addition, AT is assumed to be bounded, this conclusion still holds without the requirement that F(T] ≠ Ø. Moreover, if, in addition, E has a uniformly Gâteaux differentiable norm and is such that every closed bounded convex subset of K has the fixed point property for nonexpansive self-mappings, then the sequence {xn} converges strongly to a fixed point of T, Our iteration method is of independent interest.

AB - Let K be a nonempty closed convex subset of a real Banach space E and T be a Lipschitz pseudocontractive self-map of K with F(T) := {x ε K : T x = x} ≠ Ø. An iterative sequence {xn} is constructed for which ∥xn - Txn∥ → 0 as n → ∞. If, in addition, AT is assumed to be bounded, this conclusion still holds without the requirement that F(T] ≠ Ø. Moreover, if, in addition, E has a uniformly Gâteaux differentiable norm and is such that every closed bounded convex subset of K has the fixed point property for nonexpansive self-mappings, then the sequence {xn} converges strongly to a fixed point of T, Our iteration method is of independent interest.

UR - http://www.scopus.com/inward/record.url?scp=1442278169&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=1442278169&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:1442278169

VL - 132

SP - 831

EP - 840

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

SN - 0002-9939

IS - 3

ER -