Viscosity methods of approximation for a common fixed point of a family of quasi-nonexpansive mappings

Habtu Zegeye, Naseer Shahzad

Research output: Contribution to journalArticle

15 Citations (Scopus)

Abstract

Let K be a nonempty closed convex subset of a real reflexive Banach space E that has weakly continuous duality mapping Jφ for some gauge φ. Let Ti : K → K, i = 1, 2, ..., be a family of quasi-nonexpansive mappings with F {colon equals} ∩i ≥ 1 F (Ti) ≠ 0{combining long solidus overlay} which is a sunny nonexpansive retract of K with Q a nonexpansive retraction. For given x0 ∈ K, let {xn} be generated by the algorithm xn + 1 {colon equals} αn f (xn) + (1 - αn) Tn (xn), n ≥ 0, where f : K → K is a contraction mapping and {αn} ⊆ (0, 1) a sequence satisfying certain conditions. Suppose that {xn} satisfies condition (A). Then it is proved that {xn} converges strongly to a common fixed point over(x, ̄) = Q f (over(x, ̄)) of a family Ti, i = 1, 2, .... Moreover, over(x, ̄) is the unique solution in F to a certain variational inequality.

Original languageEnglish
Pages (from-to)2005-2012
Number of pages8
JournalNonlinear Analysis, Theory, Methods and Applications
Volume68
Issue number7
DOIs
Publication statusPublished - Apr 1 2008

Fingerprint

Viscosity Method
Nonexpansive Mapping
Common Fixed Point
Duality Mapping
Viscosity
Contraction Mapping
Retract
Retraction
Reflexive Banach Space
Approximation
Overlay
Unique Solution
Variational Inequalities
Gauge
Banach spaces
Set theory
Converge
Closed
Gages
Subset

All Science Journal Classification (ASJC) codes

  • Analysis
  • Applied Mathematics

Cite this

@article{adc02cd5e39f483f94db5643b25c673f,
title = "Viscosity methods of approximation for a common fixed point of a family of quasi-nonexpansive mappings",
abstract = "Let K be a nonempty closed convex subset of a real reflexive Banach space E that has weakly continuous duality mapping Jφ for some gauge φ. Let Ti : K → K, i = 1, 2, ..., be a family of quasi-nonexpansive mappings with F {colon equals} ∩i ≥ 1 F (Ti) ≠ 0{combining long solidus overlay} which is a sunny nonexpansive retract of K with Q a nonexpansive retraction. For given x0 ∈ K, let {xn} be generated by the algorithm xn + 1 {colon equals} αn f (xn) + (1 - αn) Tn (xn), n ≥ 0, where f : K → K is a contraction mapping and {αn} ⊆ (0, 1) a sequence satisfying certain conditions. Suppose that {xn} satisfies condition (A). Then it is proved that {xn} converges strongly to a common fixed point over(x, ̄) = Q f (over(x, ̄)) of a family Ti, i = 1, 2, .... Moreover, over(x, ̄) is the unique solution in F to a certain variational inequality.",
author = "Habtu Zegeye and Naseer Shahzad",
year = "2008",
month = "4",
day = "1",
doi = "10.1016/j.na.2007.01.027",
language = "English",
volume = "68",
pages = "2005--2012",
journal = "Nonlinear Analysis, Theory, Methods and Applications",
issn = "0362-546X",
publisher = "Elsevier Limited",
number = "7",

}

Viscosity methods of approximation for a common fixed point of a family of quasi-nonexpansive mappings. / Zegeye, Habtu; Shahzad, Naseer.

In: Nonlinear Analysis, Theory, Methods and Applications, Vol. 68, No. 7, 01.04.2008, p. 2005-2012.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Viscosity methods of approximation for a common fixed point of a family of quasi-nonexpansive mappings

AU - Zegeye, Habtu

AU - Shahzad, Naseer

PY - 2008/4/1

Y1 - 2008/4/1

N2 - Let K be a nonempty closed convex subset of a real reflexive Banach space E that has weakly continuous duality mapping Jφ for some gauge φ. Let Ti : K → K, i = 1, 2, ..., be a family of quasi-nonexpansive mappings with F {colon equals} ∩i ≥ 1 F (Ti) ≠ 0{combining long solidus overlay} which is a sunny nonexpansive retract of K with Q a nonexpansive retraction. For given x0 ∈ K, let {xn} be generated by the algorithm xn + 1 {colon equals} αn f (xn) + (1 - αn) Tn (xn), n ≥ 0, where f : K → K is a contraction mapping and {αn} ⊆ (0, 1) a sequence satisfying certain conditions. Suppose that {xn} satisfies condition (A). Then it is proved that {xn} converges strongly to a common fixed point over(x, ̄) = Q f (over(x, ̄)) of a family Ti, i = 1, 2, .... Moreover, over(x, ̄) is the unique solution in F to a certain variational inequality.

AB - Let K be a nonempty closed convex subset of a real reflexive Banach space E that has weakly continuous duality mapping Jφ for some gauge φ. Let Ti : K → K, i = 1, 2, ..., be a family of quasi-nonexpansive mappings with F {colon equals} ∩i ≥ 1 F (Ti) ≠ 0{combining long solidus overlay} which is a sunny nonexpansive retract of K with Q a nonexpansive retraction. For given x0 ∈ K, let {xn} be generated by the algorithm xn + 1 {colon equals} αn f (xn) + (1 - αn) Tn (xn), n ≥ 0, where f : K → K is a contraction mapping and {αn} ⊆ (0, 1) a sequence satisfying certain conditions. Suppose that {xn} satisfies condition (A). Then it is proved that {xn} converges strongly to a common fixed point over(x, ̄) = Q f (over(x, ̄)) of a family Ti, i = 1, 2, .... Moreover, over(x, ̄) is the unique solution in F to a certain variational inequality.

UR - http://www.scopus.com/inward/record.url?scp=38749110405&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=38749110405&partnerID=8YFLogxK

U2 - 10.1016/j.na.2007.01.027

DO - 10.1016/j.na.2007.01.027

M3 - Article

VL - 68

SP - 2005

EP - 2012

JO - Nonlinear Analysis, Theory, Methods and Applications

JF - Nonlinear Analysis, Theory, Methods and Applications

SN - 0362-546X

IS - 7

ER -