Convergence on Composite Iterative Schemes for Nonexpansive Mappings in Banach Spaces

Convergence on Composite Iterative Schemes for Nonexpansive Mappings in Banach Spaces

Let E be a reflexive Banach space with a uniformly Gâteaux differentiable norm. Suppose that every weakly compact convex subset of E has the fixed point property for nonexpansive mappings. Let C be a nonempty closed convex subset of E, f:C  →  C a contractive mapping (or...

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Autor principal: Jong Soo Jung
Formato: article
Lenguaje:EN
Publicado: SpringerOpen 2008
Materias:
Applied mathematics. Quantitative methods
T57-57.97
Analysis
QA299.6-433
Acceso en línea:https://doaj.org/article/16da234b8dba4d378fb788dfd6c77de2
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spelling oai:doaj.org-article:16da234b8dba4d378fb788dfd6c77de22021-12-02T11:29:20ZConvergence on Composite Iterative Schemes for Nonexpansive Mappings in Banach Spaces10.1155/2008/1675351687-18201687-1812https://doaj.org/article/16da234b8dba4d378fb788dfd6c77de22008-05-01T00:00:00Zhttp://dx.doi.org/10.1155/2008/167535https://doaj.org/toc/1687-1820https://doaj.org/toc/1687-1812Let E be a reflexive Banach space with a uniformly Gâteaux differentiable norm. Suppose that every weakly compact convex subset of E has the fixed point property for nonexpansive mappings. Let C be a nonempty closed convex subset of E, f:C  →  C a contractive mapping (or a weakly contractive mapping), and T:C  →  C nonexpansive mapping with the fixed point set F(T)  ≠  ∅. Let {xn} be generated by a new composite iterative scheme: yn=λnf(xn)+(1−λn)Txn, xn+1=(1−βn)yn+βnTyn, (n≥0). It is proved that {xn} converges strongly to a point in F(T), which is a solution of certain variational inequality provided that the sequence {λn}⊂(0,1) satisfies limn→∞λn=0 and ∑n=1∞λn=∞, {βn}⊂[0,a) for some 0<a<1 and the sequence {xn} is asymptotically regular.Jong Soo JungSpringerOpenarticleApplied mathematics. Quantitative methodsT57-57.97AnalysisQA299.6-433ENFixed Point Theory and Applications, Vol 2008 (2008)
institution DOAJ
collection DOAJ
language EN
topic Applied mathematics. Quantitative methods
T57-57.97
Analysis
QA299.6-433
spellingShingle Applied mathematics. Quantitative methods
T57-57.97
Analysis
QA299.6-433
Jong Soo Jung
Convergence on Composite Iterative Schemes for Nonexpansive Mappings in Banach Spaces
description Let E be a reflexive Banach space with a uniformly Gâteaux differentiable norm. Suppose that every weakly compact convex subset of E has the fixed point property for nonexpansive mappings. Let C be a nonempty closed convex subset of E, f:C  →  C a contractive mapping (or a weakly contractive mapping), and T:C  →  C nonexpansive mapping with the fixed point set F(T)  ≠  ∅. Let {xn} be generated by a new composite iterative scheme: yn=λnf(xn)+(1−λn)Txn, xn+1=(1−βn)yn+βnTyn, (n≥0). It is proved that {xn} converges strongly to a point in F(T), which is a solution of certain variational inequality provided that the sequence {λn}⊂(0,1) satisfies limn→∞λn=0 and ∑n=1∞λn=∞, {βn}⊂[0,a) for some 0<a<1 and the sequence {xn} is asymptotically regular.
format article
author Jong Soo Jung
author_facet Jong Soo Jung
author_sort Jong Soo Jung
title Convergence on Composite Iterative Schemes for Nonexpansive Mappings in Banach Spaces
title_short Convergence on Composite Iterative Schemes for Nonexpansive Mappings in Banach Spaces
title_full Convergence on Composite Iterative Schemes for Nonexpansive Mappings in Banach Spaces
title_fullStr Convergence on Composite Iterative Schemes for Nonexpansive Mappings in Banach Spaces
title_full_unstemmed Convergence on Composite Iterative Schemes for Nonexpansive Mappings in Banach Spaces
title_sort convergence on composite iterative schemes for nonexpansive mappings in banach spaces
publisher SpringerOpen
publishDate 2008
url https://doaj.org/article/16da234b8dba4d378fb788dfd6c77de2
work_keys_str_mv AT jongsoojung convergenceoncompositeiterativeschemesfornonexpansivemappingsinbanachspaces
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