Fixed point variational solutions for uniformly continuous pseudocontractions in Banach spaces

Fixed point variational solutions for uniformly continuous pseudocontractions in Banach spaces

Let E be a reflexive Banach space with a uniformly Gâteaux differentiable norm, let K be a nonempty closed convex subset of E, and let T:K→K be a uniformly continuous pseudocontraction. If f:K→K is any contraction map on K and if every nonempty closed convex and bounded subset of K has...

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Main Author: Aniefiok Udomene
Format: Article
Language:English
Published: SpringerOpen 2006-03-01
Series:Fixed Point Theory and Applications
Online Access:http://dx.doi.org/10.1155/FPTA/2006/69758
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spelling doaj-37af794a06c3414a8851a82d7d21c0212020-11-25T01:17:22ZengSpringerOpenFixed Point Theory and Applications1687-18201687-18122006-03-01200610.1155/FPTA/2006/69758Fixed point variational solutions for uniformly continuous pseudocontractions in Banach spacesAniefiok UdomeneLet E be a reflexive Banach space with a uniformly Gâteaux differentiable norm, let K be a nonempty closed convex subset of E, and let T:K→K be a uniformly continuous pseudocontraction. If f:K→K is any contraction map on K and if every nonempty closed convex and bounded subset of K has the fixed point property for nonexpansive self-mappings, then it is shown, under appropriate conditions on the sequences of real numbers {αn}, {μn}, that the iteration process z1∈K, zn+1=μn(αnTzn+(1−αn)zn)+(1−μn)f(zn), n∈ℕ, strongly converges to the fixed point of T, which is the unique solution of some variational inequality, provided that K is bounded.http://dx.doi.org/10.1155/FPTA/2006/69758
collection DOAJ
language English
format Article
sources DOAJ
author Aniefiok Udomene
spellingShingle Aniefiok Udomene
Fixed point variational solutions for uniformly continuous pseudocontractions in Banach spaces
Fixed Point Theory and Applications
author_facet Aniefiok Udomene
author_sort Aniefiok Udomene
title Fixed point variational solutions for uniformly continuous pseudocontractions in Banach spaces
title_short Fixed point variational solutions for uniformly continuous pseudocontractions in Banach spaces
title_full Fixed point variational solutions for uniformly continuous pseudocontractions in Banach spaces
title_fullStr Fixed point variational solutions for uniformly continuous pseudocontractions in Banach spaces
title_full_unstemmed Fixed point variational solutions for uniformly continuous pseudocontractions in Banach spaces
title_sort fixed point variational solutions for uniformly continuous pseudocontractions in banach spaces
publisher SpringerOpen
series Fixed Point Theory and Applications
issn 1687-1820
1687-1812
publishDate 2006-03-01
description Let E be a reflexive Banach space with a uniformly Gâteaux differentiable norm, let K be a nonempty closed convex subset of E, and let T:K→K be a uniformly continuous pseudocontraction. If f:K→K is any contraction map on K and if every nonempty closed convex and bounded subset of K has the fixed point property for nonexpansive self-mappings, then it is shown, under appropriate conditions on the sequences of real numbers {αn}, {μn}, that the iteration process z1∈K, zn+1=μn(αnTzn+(1−αn)zn)+(1−μn)f(zn), n∈ℕ, strongly converges to the fixed point of T, which is the unique solution of some variational inequality, provided that K is bounded.
url http://dx.doi.org/10.1155/FPTA/2006/69758
work_keys_str_mv AT aniefiokudomene fixedpointvariationalsolutionsforuniformlycontinuouspseudocontractionsinbanachspaces
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