Approximate Proximal Point Algorithms for Finding Zeroes of Maximal Monotone Operators in Hilbert Spaces
<p>Abstract</p> <p>Let <inline-formula> <graphic file="1029-242X-2008-598191-i1.gif"/></inline-formula> be a real Hilbert space, <inline-formula> <graphic file="1029-242X-2008-598191-i2.gif"/></inline-formula> a nonempty close...
Main Authors: | , , |
---|---|
Format: | Article |
Language: | English |
Published: |
SpringerOpen
2008-01-01
|
Series: | Journal of Inequalities and Applications |
Online Access: | http://www.journalofinequalitiesandapplications.com/content/2008/598191 |
id |
doaj-8e5055a292214ff29e6ef730f2a1432d |
---|---|
record_format |
Article |
spelling |
doaj-8e5055a292214ff29e6ef730f2a1432d2020-11-25T01:03:38ZengSpringerOpenJournal of Inequalities and Applications1025-58341029-242X2008-01-0120081598191Approximate Proximal Point Algorithms for Finding Zeroes of Maximal Monotone Operators in Hilbert SpacesKang ShinMinZhou HaiyunCho YeolJe<p>Abstract</p> <p>Let <inline-formula> <graphic file="1029-242X-2008-598191-i1.gif"/></inline-formula> be a real Hilbert space, <inline-formula> <graphic file="1029-242X-2008-598191-i2.gif"/></inline-formula> a nonempty closed convex subset of <inline-formula> <graphic file="1029-242X-2008-598191-i3.gif"/></inline-formula>, and <inline-formula> <graphic file="1029-242X-2008-598191-i4.gif"/></inline-formula> a maximal monotone operator with <inline-formula> <graphic file="1029-242X-2008-598191-i5.gif"/></inline-formula>. Let <inline-formula> <graphic file="1029-242X-2008-598191-i6.gif"/></inline-formula> be the metric projection of <inline-formula> <graphic file="1029-242X-2008-598191-i7.gif"/></inline-formula> onto <inline-formula> <graphic file="1029-242X-2008-598191-i8.gif"/></inline-formula>. Suppose that, for any given <inline-formula> <graphic file="1029-242X-2008-598191-i9.gif"/></inline-formula>, <inline-formula> <graphic file="1029-242X-2008-598191-i10.gif"/></inline-formula>, and <inline-formula> <graphic file="1029-242X-2008-598191-i11.gif"/></inline-formula>, there exists <inline-formula> <graphic file="1029-242X-2008-598191-i12.gif"/></inline-formula> satisfying the following set-valued mapping equation: <inline-formula> <graphic file="1029-242X-2008-598191-i13.gif"/></inline-formula> for all <inline-formula> <graphic file="1029-242X-2008-598191-i14.gif"/></inline-formula>, where <inline-formula> <graphic file="1029-242X-2008-598191-i15.gif"/></inline-formula> with <inline-formula> <graphic file="1029-242X-2008-598191-i16.gif"/></inline-formula> as <inline-formula> <graphic file="1029-242X-2008-598191-i17.gif"/></inline-formula> and <inline-formula> <graphic file="1029-242X-2008-598191-i18.gif"/></inline-formula> is regarded as an error sequence such that <inline-formula> <graphic file="1029-242X-2008-598191-i19.gif"/></inline-formula>. Let <inline-formula> <graphic file="1029-242X-2008-598191-i20.gif"/></inline-formula> be a real sequence such that <inline-formula> <graphic file="1029-242X-2008-598191-i21.gif"/></inline-formula> as <inline-formula> <graphic file="1029-242X-2008-598191-i22.gif"/></inline-formula> and <inline-formula> <graphic file="1029-242X-2008-598191-i23.gif"/></inline-formula>. For any fixed <inline-formula> <graphic file="1029-242X-2008-598191-i24.gif"/></inline-formula>, define a sequence <inline-formula> <graphic file="1029-242X-2008-598191-i25.gif"/></inline-formula> iteratively as <inline-formula> <graphic file="1029-242X-2008-598191-i26.gif"/></inline-formula> for all <inline-formula> <graphic file="1029-242X-2008-598191-i27.gif"/></inline-formula>. Then <inline-formula> <graphic file="1029-242X-2008-598191-i28.gif"/></inline-formula> converges strongly to a point <inline-formula> <graphic file="1029-242X-2008-598191-i29.gif"/></inline-formula> as <inline-formula> <graphic file="1029-242X-2008-598191-i30.gif"/></inline-formula>, where <inline-formula> <graphic file="1029-242X-2008-598191-i31.gif"/></inline-formula><inline-formula> <graphic file="1029-242X-2008-598191-i32.gif"/></inline-formula>.</p>http://www.journalofinequalitiesandapplications.com/content/2008/598191 |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Kang ShinMin Zhou Haiyun Cho YeolJe |
spellingShingle |
Kang ShinMin Zhou Haiyun Cho YeolJe Approximate Proximal Point Algorithms for Finding Zeroes of Maximal Monotone Operators in Hilbert Spaces Journal of Inequalities and Applications |
author_facet |
Kang ShinMin Zhou Haiyun Cho YeolJe |
author_sort |
Kang ShinMin |
title |
Approximate Proximal Point Algorithms for Finding Zeroes of Maximal Monotone Operators in Hilbert Spaces |
title_short |
Approximate Proximal Point Algorithms for Finding Zeroes of Maximal Monotone Operators in Hilbert Spaces |
title_full |
Approximate Proximal Point Algorithms for Finding Zeroes of Maximal Monotone Operators in Hilbert Spaces |
title_fullStr |
Approximate Proximal Point Algorithms for Finding Zeroes of Maximal Monotone Operators in Hilbert Spaces |
title_full_unstemmed |
Approximate Proximal Point Algorithms for Finding Zeroes of Maximal Monotone Operators in Hilbert Spaces |
title_sort |
approximate proximal point algorithms for finding zeroes of maximal monotone operators in hilbert spaces |
publisher |
SpringerOpen |
series |
Journal of Inequalities and Applications |
issn |
1025-5834 1029-242X |
publishDate |
2008-01-01 |
description |
<p>Abstract</p> <p>Let <inline-formula> <graphic file="1029-242X-2008-598191-i1.gif"/></inline-formula> be a real Hilbert space, <inline-formula> <graphic file="1029-242X-2008-598191-i2.gif"/></inline-formula> a nonempty closed convex subset of <inline-formula> <graphic file="1029-242X-2008-598191-i3.gif"/></inline-formula>, and <inline-formula> <graphic file="1029-242X-2008-598191-i4.gif"/></inline-formula> a maximal monotone operator with <inline-formula> <graphic file="1029-242X-2008-598191-i5.gif"/></inline-formula>. Let <inline-formula> <graphic file="1029-242X-2008-598191-i6.gif"/></inline-formula> be the metric projection of <inline-formula> <graphic file="1029-242X-2008-598191-i7.gif"/></inline-formula> onto <inline-formula> <graphic file="1029-242X-2008-598191-i8.gif"/></inline-formula>. Suppose that, for any given <inline-formula> <graphic file="1029-242X-2008-598191-i9.gif"/></inline-formula>, <inline-formula> <graphic file="1029-242X-2008-598191-i10.gif"/></inline-formula>, and <inline-formula> <graphic file="1029-242X-2008-598191-i11.gif"/></inline-formula>, there exists <inline-formula> <graphic file="1029-242X-2008-598191-i12.gif"/></inline-formula> satisfying the following set-valued mapping equation: <inline-formula> <graphic file="1029-242X-2008-598191-i13.gif"/></inline-formula> for all <inline-formula> <graphic file="1029-242X-2008-598191-i14.gif"/></inline-formula>, where <inline-formula> <graphic file="1029-242X-2008-598191-i15.gif"/></inline-formula> with <inline-formula> <graphic file="1029-242X-2008-598191-i16.gif"/></inline-formula> as <inline-formula> <graphic file="1029-242X-2008-598191-i17.gif"/></inline-formula> and <inline-formula> <graphic file="1029-242X-2008-598191-i18.gif"/></inline-formula> is regarded as an error sequence such that <inline-formula> <graphic file="1029-242X-2008-598191-i19.gif"/></inline-formula>. Let <inline-formula> <graphic file="1029-242X-2008-598191-i20.gif"/></inline-formula> be a real sequence such that <inline-formula> <graphic file="1029-242X-2008-598191-i21.gif"/></inline-formula> as <inline-formula> <graphic file="1029-242X-2008-598191-i22.gif"/></inline-formula> and <inline-formula> <graphic file="1029-242X-2008-598191-i23.gif"/></inline-formula>. For any fixed <inline-formula> <graphic file="1029-242X-2008-598191-i24.gif"/></inline-formula>, define a sequence <inline-formula> <graphic file="1029-242X-2008-598191-i25.gif"/></inline-formula> iteratively as <inline-formula> <graphic file="1029-242X-2008-598191-i26.gif"/></inline-formula> for all <inline-formula> <graphic file="1029-242X-2008-598191-i27.gif"/></inline-formula>. Then <inline-formula> <graphic file="1029-242X-2008-598191-i28.gif"/></inline-formula> converges strongly to a point <inline-formula> <graphic file="1029-242X-2008-598191-i29.gif"/></inline-formula> as <inline-formula> <graphic file="1029-242X-2008-598191-i30.gif"/></inline-formula>, where <inline-formula> <graphic file="1029-242X-2008-598191-i31.gif"/></inline-formula><inline-formula> <graphic file="1029-242X-2008-598191-i32.gif"/></inline-formula>.</p> |
url |
http://www.journalofinequalitiesandapplications.com/content/2008/598191 |
work_keys_str_mv |
AT kangshinmin approximateproximalpointalgorithmsforfindingzeroesofmaximalmonotoneoperatorsinhilbertspaces AT zhouhaiyun approximateproximalpointalgorithmsforfindingzeroesofmaximalmonotoneoperatorsinhilbertspaces AT choyeolje approximateproximalpointalgorithmsforfindingzeroesofmaximalmonotoneoperatorsinhilbertspaces |
_version_ |
1725200185996345344 |