Approximate fixed point sequences and convergence theorems for asymptotically pseudocontractive mappings

C. E. Chidume, Hab Zegeye

Research output: Contribution to journalReview article

9 Citations (Scopus)

Abstract

Let K be a nonempty closed convex and bounded subset of a real Banach space E and T : K → K be uniformly L-Lipschitzian, uniformly asymptotically regular with sequence {εn}, and asymptotically pseudocontractive with constant {kn}, where {kn} and {εn} satisfy certain mild conditions. Let a sequence {xn} be generated from x1 ∈ K by xn+1 := (1 - λn)xn + λnTnxn - λnθn(xn - x1), for all integers n ≥ 1, where {λn} and {θn} are real sequences satisfying appropriate conditions, then ∥xn - Txn∥ → 0 as n → ∞. Moreover, if E is reflexive, and has uniform normal structure with coefficient N(E) and L < N(E)1/2 and has a uniformly Gâteaux differentiable norm, and T satisfies an additional mild condition, then {xn} also converges strongly to a fixed point of T.

Original languageEnglish
Pages (from-to)354-366
Number of pages13
JournalJournal of Mathematical Analysis and Applications
Volume278
Issue number2
DOIs
Publication statusPublished - Feb 15 2003

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Approximate Fixed Point
Pseudocontractive Mapping
Banach spaces
Set theory
Convergence Theorem
Uniform Normal Structure
Differentiable
Fixed point
Banach space
Converge
Norm
Closed
Integer
Subset
Coefficient

All Science Journal Classification (ASJC) codes

  • Analysis
  • Applied Mathematics

Cite this

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title = "Approximate fixed point sequences and convergence theorems for asymptotically pseudocontractive mappings",
abstract = "Let K be a nonempty closed convex and bounded subset of a real Banach space E and T : K → K be uniformly L-Lipschitzian, uniformly asymptotically regular with sequence {εn}, and asymptotically pseudocontractive with constant {kn}, where {kn} and {εn} satisfy certain mild conditions. Let a sequence {xn} be generated from x1 ∈ K by xn+1 := (1 - λn)xn + λnTnxn - λnθn(xn - x1), for all integers n ≥ 1, where {λn} and {θn} are real sequences satisfying appropriate conditions, then ∥xn - Txn∥ → 0 as n → ∞. Moreover, if E is reflexive, and has uniform normal structure with coefficient N(E) and L < N(E)1/2 and has a uniformly G{\^a}teaux differentiable norm, and T satisfies an additional mild condition, then {xn} also converges strongly to a fixed point of T.",
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Approximate fixed point sequences and convergence theorems for asymptotically pseudocontractive mappings. / Chidume, C. E.; Zegeye, Hab.

In: Journal of Mathematical Analysis and Applications, Vol. 278, No. 2, 15.02.2003, p. 354-366.

Research output: Contribution to journalReview article

TY - JOUR

T1 - Approximate fixed point sequences and convergence theorems for asymptotically pseudocontractive mappings

AU - Chidume, C. E.

AU - Zegeye, Hab

PY - 2003/2/15

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N2 - Let K be a nonempty closed convex and bounded subset of a real Banach space E and T : K → K be uniformly L-Lipschitzian, uniformly asymptotically regular with sequence {εn}, and asymptotically pseudocontractive with constant {kn}, where {kn} and {εn} satisfy certain mild conditions. Let a sequence {xn} be generated from x1 ∈ K by xn+1 := (1 - λn)xn + λnTnxn - λnθn(xn - x1), for all integers n ≥ 1, where {λn} and {θn} are real sequences satisfying appropriate conditions, then ∥xn - Txn∥ → 0 as n → ∞. Moreover, if E is reflexive, and has uniform normal structure with coefficient N(E) and L < N(E)1/2 and has a uniformly Gâteaux differentiable norm, and T satisfies an additional mild condition, then {xn} also converges strongly to a fixed point of T.

AB - Let K be a nonempty closed convex and bounded subset of a real Banach space E and T : K → K be uniformly L-Lipschitzian, uniformly asymptotically regular with sequence {εn}, and asymptotically pseudocontractive with constant {kn}, where {kn} and {εn} satisfy certain mild conditions. Let a sequence {xn} be generated from x1 ∈ K by xn+1 := (1 - λn)xn + λnTnxn - λnθn(xn - x1), for all integers n ≥ 1, where {λn} and {θn} are real sequences satisfying appropriate conditions, then ∥xn - Txn∥ → 0 as n → ∞. Moreover, if E is reflexive, and has uniform normal structure with coefficient N(E) and L < N(E)1/2 and has a uniformly Gâteaux differentiable norm, and T satisfies an additional mild condition, then {xn} also converges strongly to a fixed point of T.

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