Approximation methods for nonlinear operator equations

C. E. Chidume, H. Zegeye

Research output: Contribution to journalArticle

22 Citations (Scopus)

Abstract

Let E be a real normed linear space and A : E → E be a uniformly quasi-accretive map. For arbitrary x1 ∈ E define the sequence xn ∈ E by xn+i := xn - αnAxn, n ≥1, where {αn} is a positve real sequence satisfying the following conditions: (i) Σαn = ∞; (ii) lim αn = 0. For x* ∈ N (A) := {x ∈ E : Ax = 0}, assume that σ := infn∈N0 ψ(∥xn+1-x*∥)/∥xn+1-x*∥ > 0 and that ∥Axn+1 - Axn∥ → 0, where N0 := {n ∈ N (the set of all positive integers): xn+1 ≠ x*} and ψ : [0,∞) → [0,∞) is a strictly increasing function with ψ(0) = 0. It is proved that a Mann-type iteration process converges strongly to x*. Furthermore if, in addition, A is a uniformly continuous map, it is proved, without the condition on σ, that the Mann-type iteration process converges strongly to x*. As a consequence, corresponding convergence theorems for fixed points of hemi-contractive maps are proved.

Original languageEnglish
Pages (from-to)2467-2478
Number of pages12
JournalProceedings of the American Mathematical Society
Volume131
Issue number8
DOIs
Publication statusPublished - Aug 1 2003

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Nonlinear Operator Equations
Approximation Methods
Converge
Iteration
Normed Linear Space
Uniformly continuous
Increasing Functions
Continuous Map
Convergence Theorem
Strictly
Fixed point
Integer
Arbitrary

All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Applied Mathematics

Cite this

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abstract = "Let E be a real normed linear space and A : E → E be a uniformly quasi-accretive map. For arbitrary x1 ∈ E define the sequence xn ∈ E by xn+i := xn - αnAxn, n ≥1, where {αn} is a positve real sequence satisfying the following conditions: (i) Σαn = ∞; (ii) lim αn = 0. For x* ∈ N (A) := {x ∈ E : Ax = 0}, assume that σ := infn∈N0 ψ(∥xn+1-x*∥)/∥xn+1-x*∥ > 0 and that ∥Axn+1 - Axn∥ → 0, where N0 := {n ∈ N (the set of all positive integers): xn+1 ≠ x*} and ψ : [0,∞) → [0,∞) is a strictly increasing function with ψ(0) = 0. It is proved that a Mann-type iteration process converges strongly to x*. Furthermore if, in addition, A is a uniformly continuous map, it is proved, without the condition on σ, that the Mann-type iteration process converges strongly to x*. As a consequence, corresponding convergence theorems for fixed points of hemi-contractive maps are proved.",
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Approximation methods for nonlinear operator equations. / Chidume, C. E.; Zegeye, H.

In: Proceedings of the American Mathematical Society, Vol. 131, No. 8, 01.08.2003, p. 2467-2478.

Research output: Contribution to journalArticle

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