Convergence on Composite Iterative Schemes for Nonexpansive Mappings in Banach Spaces

Fixed Point Theory and Applications. 2008;2008(1):167535

 

Journal Homepage

Journal Title: Fixed Point Theory and Applications

ISSN: 1687-1820 (Print); 1687-1812 (Online)

Publisher: SpringerOpen

LCC Subject Category: Technology: Technology (General): Industrial engineering. Management engineering: Applied mathematics. Quantitative methods | Science: Mathematics: Analysis

Country of publisher: United Kingdom

Language of fulltext: English

Full-text formats available: PDF, HTML

 

AUTHORS

Jung JongSoo

EDITORIAL INFORMATION

Blind peer review

Editorial Board

Instructions for authors

Time From Submission to Publication: 13 weeks

 

Abstract | Full Text

<p>Abstract</p> <p>Let <inline-formula> <graphic file="1687-1812-2008-167535-i1.gif"/></inline-formula> be a reflexive Banach space with a uniformly G&#226;teaux differentiable norm. Suppose that every weakly compact convex subset of <inline-formula> <graphic file="1687-1812-2008-167535-i2.gif"/></inline-formula> has the fixed point property for nonexpansive mappings. Let <inline-formula> <graphic file="1687-1812-2008-167535-i3.gif"/></inline-formula> be a nonempty closed convex subset of <inline-formula> <graphic file="1687-1812-2008-167535-i4.gif"/></inline-formula>, <inline-formula> <graphic file="1687-1812-2008-167535-i5.gif"/></inline-formula> a contractive mapping (or a weakly contractive mapping), and <inline-formula> <graphic file="1687-1812-2008-167535-i6.gif"/></inline-formula> nonexpansive mapping with the fixed point set <inline-formula> <graphic file="1687-1812-2008-167535-i7.gif"/></inline-formula>. Let <inline-formula> <graphic file="1687-1812-2008-167535-i8.gif"/></inline-formula> be generated by a new composite iterative scheme: <inline-formula> <graphic file="1687-1812-2008-167535-i9.gif"/></inline-formula>, <inline-formula> <graphic file="1687-1812-2008-167535-i10.gif"/></inline-formula>, <inline-formula> <graphic file="1687-1812-2008-167535-i11.gif"/></inline-formula>. It is proved that <inline-formula> <graphic file="1687-1812-2008-167535-i12.gif"/></inline-formula> converges strongly to a point in <inline-formula> <graphic file="1687-1812-2008-167535-i13.gif"/></inline-formula>, which is a solution of certain variational inequality provided that the sequence <inline-formula> <graphic file="1687-1812-2008-167535-i14.gif"/></inline-formula> satisfies <inline-formula> <graphic file="1687-1812-2008-167535-i15.gif"/></inline-formula> and <inline-formula> <graphic file="1687-1812-2008-167535-i16.gif"/></inline-formula>, <inline-formula> <graphic file="1687-1812-2008-167535-i17.gif"/></inline-formula> for some <inline-formula> <graphic file="1687-1812-2008-167535-i18.gif"/></inline-formula> and the sequence <inline-formula> <graphic file="1687-1812-2008-167535-i19.gif"/></inline-formula> is asymptotically regular.</p>