Convergence theorems for ψ-expansive and accretive mappings

Habtu Zegeye, Naseer Shahzad

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

Let E be a real Banach space, and let A : D (A) ⊆ E → E be a Lipschitz, ψ-expansive and accretive mapping such that over(c o, -) (D (A)) ⊆ ∩λ > 0 R (I + λ A). Suppose that there exists x0 ∈ D (A), where one of the following holds: (i) There exists R > 0 such that ψ (R) > 2 {norm of matrix} A (x0) {norm of matrix}; or (ii) There exists a bounded neighborhood U of x0 such that t (x - x0) ∉ A x for x ∈ ∂ U ∩ D (A) and t < 0. An iterative sequence {xn} is constructed to converge strongly to a zero of A. Related results deal with the strong convergence of this iteration process to fixed points of ψ-expansive and pseudocontractive mappings in real Banach spaces. The convergence results established in this paper are new for this more general class of ψ-expansive and accretive or pseudocontractive mappings.

Original languageEnglish
Pages (from-to)73-82
Number of pages10
JournalNonlinear Analysis, Theory, Methods and Applications
Volume66
Issue number1
DOIs
Publication statusPublished - Jan 1 2007

Fingerprint

Accretive Mapping
Pseudocontractive Mapping
Convergence Theorem
Banach spaces
Banach space
Norm
Strong Convergence
Convergence Results
Lipschitz
Fixed point
Converge
Iteration
Zero
Class

All Science Journal Classification (ASJC) codes

  • Analysis
  • Applied Mathematics

Cite this

@article{ad24c199d66e4fc29c700671602b1aeb,
title = "Convergence theorems for ψ-expansive and accretive mappings",
abstract = "Let E be a real Banach space, and let A : D (A) ⊆ E → E be a Lipschitz, ψ-expansive and accretive mapping such that over(c o, -) (D (A)) ⊆ ∩λ > 0 R (I + λ A). Suppose that there exists x0 ∈ D (A), where one of the following holds: (i) There exists R > 0 such that ψ (R) > 2 {norm of matrix} A (x0) {norm of matrix}; or (ii) There exists a bounded neighborhood U of x0 such that t (x - x0) ∉ A x for x ∈ ∂ U ∩ D (A) and t < 0. An iterative sequence {xn} is constructed to converge strongly to a zero of A. Related results deal with the strong convergence of this iteration process to fixed points of ψ-expansive and pseudocontractive mappings in real Banach spaces. The convergence results established in this paper are new for this more general class of ψ-expansive and accretive or pseudocontractive mappings.",
author = "Habtu Zegeye and Naseer Shahzad",
year = "2007",
month = "1",
day = "1",
doi = "10.1016/j.na.2005.11.011",
language = "English",
volume = "66",
pages = "73--82",
journal = "Nonlinear Analysis, Theory, Methods and Applications",
issn = "0362-546X",
publisher = "Elsevier Limited",
number = "1",

}

Convergence theorems for ψ-expansive and accretive mappings. / Zegeye, Habtu; Shahzad, Naseer.

In: Nonlinear Analysis, Theory, Methods and Applications, Vol. 66, No. 1, 01.01.2007, p. 73-82.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Convergence theorems for ψ-expansive and accretive mappings

AU - Zegeye, Habtu

AU - Shahzad, Naseer

PY - 2007/1/1

Y1 - 2007/1/1

N2 - Let E be a real Banach space, and let A : D (A) ⊆ E → E be a Lipschitz, ψ-expansive and accretive mapping such that over(c o, -) (D (A)) ⊆ ∩λ > 0 R (I + λ A). Suppose that there exists x0 ∈ D (A), where one of the following holds: (i) There exists R > 0 such that ψ (R) > 2 {norm of matrix} A (x0) {norm of matrix}; or (ii) There exists a bounded neighborhood U of x0 such that t (x - x0) ∉ A x for x ∈ ∂ U ∩ D (A) and t < 0. An iterative sequence {xn} is constructed to converge strongly to a zero of A. Related results deal with the strong convergence of this iteration process to fixed points of ψ-expansive and pseudocontractive mappings in real Banach spaces. The convergence results established in this paper are new for this more general class of ψ-expansive and accretive or pseudocontractive mappings.

AB - Let E be a real Banach space, and let A : D (A) ⊆ E → E be a Lipschitz, ψ-expansive and accretive mapping such that over(c o, -) (D (A)) ⊆ ∩λ > 0 R (I + λ A). Suppose that there exists x0 ∈ D (A), where one of the following holds: (i) There exists R > 0 such that ψ (R) > 2 {norm of matrix} A (x0) {norm of matrix}; or (ii) There exists a bounded neighborhood U of x0 such that t (x - x0) ∉ A x for x ∈ ∂ U ∩ D (A) and t < 0. An iterative sequence {xn} is constructed to converge strongly to a zero of A. Related results deal with the strong convergence of this iteration process to fixed points of ψ-expansive and pseudocontractive mappings in real Banach spaces. The convergence results established in this paper are new for this more general class of ψ-expansive and accretive or pseudocontractive mappings.

UR - http://www.scopus.com/inward/record.url?scp=33750622331&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33750622331&partnerID=8YFLogxK

U2 - 10.1016/j.na.2005.11.011

DO - 10.1016/j.na.2005.11.011

M3 - Article

VL - 66

SP - 73

EP - 82

JO - Nonlinear Analysis, Theory, Methods and Applications

JF - Nonlinear Analysis, Theory, Methods and Applications

SN - 0362-546X

IS - 1

ER -