Fixed Point Theorems for an Elastic Nonlinear Mapping in Banach Spaces

Fixed Point Theorems for an Elastic Nonlinear Mapping in Banach Spaces

Let E be a smooth Banach space with a norm ·. Let V(x,y)=x2+y2-2 x,Jy for any x,y∈E, where ·,· stands for the duality pair and J is the normalized duality mapping. We define a V-strongly nonexpansive mapping by V(·,·). This nonlinear mapping is nonexpansive in a Hilbert space. However, we show that...

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Main Author: Hiroko Manaka
Format: Article
Language:English
Published: Hindawi Limited 2015-01-01
Series:Abstract and Applied Analysis
Online Access:http://dx.doi.org/10.1155/2015/760671
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spelling doaj-f157823c164d47ef9143846e61a4b84d2020-11-24T21:29:57ZengHindawi LimitedAbstract and Applied Analysis1085-33751687-04092015-01-01201510.1155/2015/760671760671Fixed Point Theorems for an Elastic Nonlinear Mapping in Banach SpacesHiroko Manaka0Department of Mathematics, Graduate School of Environment and Information Sciences, Yokohama National University, Tokiwadai, Hodogayaku, Yokohama 240-8501, JapanLet E be a smooth Banach space with a norm ·. Let V(x,y)=x2+y2-2 x,Jy for any x,y∈E, where ·,· stands for the duality pair and J is the normalized duality mapping. We define a V-strongly nonexpansive mapping by V(·,·). This nonlinear mapping is nonexpansive in a Hilbert space. However, we show that there exists a V-strongly nonexpansive mapping with fixed points which is not nonexpansive in a Banach space. In this paper, we show a weak convergence theorem and strong convergence theorems for fixed points of this elastic nonlinear mapping and give the existence theorem.http://dx.doi.org/10.1155/2015/760671
collection DOAJ
language English
format Article
sources DOAJ
author Hiroko Manaka
spellingShingle Hiroko Manaka
Fixed Point Theorems for an Elastic Nonlinear Mapping in Banach Spaces
Abstract and Applied Analysis
author_facet Hiroko Manaka
author_sort Hiroko Manaka
title Fixed Point Theorems for an Elastic Nonlinear Mapping in Banach Spaces
title_short Fixed Point Theorems for an Elastic Nonlinear Mapping in Banach Spaces
title_full Fixed Point Theorems for an Elastic Nonlinear Mapping in Banach Spaces
title_fullStr Fixed Point Theorems for an Elastic Nonlinear Mapping in Banach Spaces
title_full_unstemmed Fixed Point Theorems for an Elastic Nonlinear Mapping in Banach Spaces
title_sort fixed point theorems for an elastic nonlinear mapping in banach spaces
publisher Hindawi Limited
series Abstract and Applied Analysis
issn 1085-3375
1687-0409
publishDate 2015-01-01
description Let E be a smooth Banach space with a norm ·. Let V(x,y)=x2+y2-2 x,Jy for any x,y∈E, where ·,· stands for the duality pair and J is the normalized duality mapping. We define a V-strongly nonexpansive mapping by V(·,·). This nonlinear mapping is nonexpansive in a Hilbert space. However, we show that there exists a V-strongly nonexpansive mapping with fixed points which is not nonexpansive in a Banach space. In this paper, we show a weak convergence theorem and strong convergence theorems for fixed points of this elastic nonlinear mapping and give the existence theorem.
url http://dx.doi.org/10.1155/2015/760671
work_keys_str_mv AT hirokomanaka fixedpointtheoremsforanelasticnonlinearmappinginbanachspaces
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