Open Access

Approximate Proximal Point Algorithms for Finding Zeroes of Maximal Monotone Operators in Hilbert Spaces

Journal of Inequalities and Applications20072008:598191

DOI: 10.1155/2008/598191

Received: 1 March 2007

Accepted: 27 November 2007

Published: 4 December 2007

Abstract

Let https://static-content.springer.com/image/art%3A10.1155%2F2008%2F598191/MediaObjects/13660_2007_Article_1832_IEq1_HTML.gif be a real Hilbert space, https://static-content.springer.com/image/art%3A10.1155%2F2008%2F598191/MediaObjects/13660_2007_Article_1832_IEq2_HTML.gif a nonempty closed convex subset of https://static-content.springer.com/image/art%3A10.1155%2F2008%2F598191/MediaObjects/13660_2007_Article_1832_IEq3_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F598191/MediaObjects/13660_2007_Article_1832_IEq4_HTML.gif a maximal monotone operator with https://static-content.springer.com/image/art%3A10.1155%2F2008%2F598191/MediaObjects/13660_2007_Article_1832_IEq5_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2008%2F598191/MediaObjects/13660_2007_Article_1832_IEq6_HTML.gif be the metric projection of https://static-content.springer.com/image/art%3A10.1155%2F2008%2F598191/MediaObjects/13660_2007_Article_1832_IEq7_HTML.gif onto https://static-content.springer.com/image/art%3A10.1155%2F2008%2F598191/MediaObjects/13660_2007_Article_1832_IEq8_HTML.gif . Suppose that, for any given https://static-content.springer.com/image/art%3A10.1155%2F2008%2F598191/MediaObjects/13660_2007_Article_1832_IEq9_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2008%2F598191/MediaObjects/13660_2007_Article_1832_IEq10_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F598191/MediaObjects/13660_2007_Article_1832_IEq11_HTML.gif , there exists https://static-content.springer.com/image/art%3A10.1155%2F2008%2F598191/MediaObjects/13660_2007_Article_1832_IEq12_HTML.gif satisfying the following set-valued mapping equation: https://static-content.springer.com/image/art%3A10.1155%2F2008%2F598191/MediaObjects/13660_2007_Article_1832_IEq13_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2008%2F598191/MediaObjects/13660_2007_Article_1832_IEq14_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2008%2F598191/MediaObjects/13660_2007_Article_1832_IEq15_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2008%2F598191/MediaObjects/13660_2007_Article_1832_IEq16_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2008%2F598191/MediaObjects/13660_2007_Article_1832_IEq17_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F598191/MediaObjects/13660_2007_Article_1832_IEq18_HTML.gif is regarded as an error sequence such that https://static-content.springer.com/image/art%3A10.1155%2F2008%2F598191/MediaObjects/13660_2007_Article_1832_IEq19_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2008%2F598191/MediaObjects/13660_2007_Article_1832_IEq20_HTML.gif be a real sequence such that https://static-content.springer.com/image/art%3A10.1155%2F2008%2F598191/MediaObjects/13660_2007_Article_1832_IEq21_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2008%2F598191/MediaObjects/13660_2007_Article_1832_IEq22_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F598191/MediaObjects/13660_2007_Article_1832_IEq23_HTML.gif . For any fixed https://static-content.springer.com/image/art%3A10.1155%2F2008%2F598191/MediaObjects/13660_2007_Article_1832_IEq24_HTML.gif , define a sequence https://static-content.springer.com/image/art%3A10.1155%2F2008%2F598191/MediaObjects/13660_2007_Article_1832_IEq25_HTML.gif iteratively as https://static-content.springer.com/image/art%3A10.1155%2F2008%2F598191/MediaObjects/13660_2007_Article_1832_IEq26_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2008%2F598191/MediaObjects/13660_2007_Article_1832_IEq27_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2008%2F598191/MediaObjects/13660_2007_Article_1832_IEq28_HTML.gif converges strongly to a point https://static-content.springer.com/image/art%3A10.1155%2F2008%2F598191/MediaObjects/13660_2007_Article_1832_IEq29_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2008%2F598191/MediaObjects/13660_2007_Article_1832_IEq30_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2008%2F598191/MediaObjects/13660_2007_Article_1832_IEq31_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2008%2F598191/MediaObjects/13660_2007_Article_1832_IEq32_HTML.gif .

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Authors’ Affiliations

(1)
Department of Mathematics Education and the RINS, Gyeongsang National University
(2)
Department of Mathematics and the RINS, Gyeongsang National University
(3)
Department of Mathematics, Shijiazhuang Mechanical Engineering College

Copyright

© Yeol Je Cho et al. 2008

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.