Báo cáo hóa học: Research Article The Over-Relaxed A-Proximal Point Algorithm for General Nonlinear Mixed Set-Valued Inclusion Framework

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Báo cáo hóa học: " Research Article The Over-Relaxed A-Proximal Point Algorithm for General Nonlinear Mixed Set-Valued Inclusion Framework"Hindawi Publishing CorporationFixed Point Theory and ApplicationsVolume 2011, Article ID 840978, 12 pagesdoi:10.1155/2011/840978Research ArticleThe Over-Relaxed A-Proximal PointAlgorithm for General Nonlinear Mixed Set-ValuedInclusion Framework Xian Bing Pan,1 Hong Gang Li,2 and An Jian Xu3 1 Yitong College, Chongqing University of Posts and Telecommunications, Chongqing 400065, China 2 Institute of Applied Mathematics Research, Chongqing University of Posts and Telecommunications, Chongqing 400065, China 3 College of Mathematics and Statistics, Chongqing University of Technology, Chongqing 400054, China Correspondence should be addressed to Xian Bing Pan, panxianb@163.com Received 16 November 2010; Accepted 10 January 2011 Academic Editor: T. Benavides Copyright q 2011 Xian Bing Pan et al. 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. The purpose of this paper is 1 a general nonlinear mixed set-valued inclusion framework for the over-relaxed A-proximal point algorithm based on the A, η -accretive mapping is introduced, and 2 it is applied to the approximation solvability of a general class of inclusions problems using the generalized resolvent operator technique due to Lan-Cho-Verma, and the convergence of iterative sequences generated by the algorithm is discussed in q-uniformly smooth Banach spaces. The results presented in the paper improve and extend some known results in the literature.1. IntroductionIn recent years, various set-valued variational inclusion frameworks, which have wideapplications to many fields including, for example, mechanics, physics, optimization andcontrol, nonlinear programming, economics, and engineering sciences have been intensivelystudied by Ding and Luo 1 , Verma 2 , Huang 3 , Fang and Huang 4 , Fang et al. 5 ,Lan et al. 6 , Zhang et al. 7 , respectively. Recently, Verma 8 has intended to developa general inclusion framework for the over-relaxed A-proximal point algorithm 9 basedon the A-maximal monotonicity. In 2007-2008, Li 10, 11 has studied the algorithm for anew class of generalized nonlinear fuzzy set-valued variational inclusions involving H, η -monotone mappings and an existence theorem of solutions for the variational inclusions,and a new iterative algorithm 12 for a new class of general nonlinear fuzzy mulitvaluedquasivariational inclusions involving G, η -monotone mappings in Hilbert spaces, anddiscussed a new perturbed Ishikawa iterative algorithm for nonlinear mixed set-valued2 Fixed Point Theory and Applicationsquasivariational inclusions involving A, η -accretive mappings, the stability 13 and theconvergence of the iterative sequences in q-uniformly smooth Banach spaces by using theresolvent operator technique due to Lan et al. 6 . Inspired and motivated by recent research work in this field, in this paper, a generalnonlinear mixed set-valued inclusion framework for the over-relaxed A-proximal pointalgorithm based on the A, η -accretive mapping is introduced, which is applied to theapproximation solvability of a general class of inclusions problems by the generalizedresolvent operator technique, and the convergence of iterative sequences generated bythe algorithm is discussed in q-uniformly smooth Banach spaces. For more literature, werecommend to the reader 1–17 .2. PreliminariesLet X be a real Banach space with dual space X ∗ , and let ·, · be the dual pair between Xand X ∗ , let 2X denote the family of all the nonempty subsets of X , and let CB X denotethe family of all nonempty closed bounded subsets of X . The generalized duality mapping ∗Jq : X → 2X is single-valued if X ∗ is strictly convex 14 , or X is uniformly smooth space. Inwhat follows we always denote the single-valued generalized duality mapping by Jq in realuniformly smooth Banach space X unless otherwise stated. We consider the following generalnonlinear mixed set-valued inclusion problem with A, η -accretive mappings (GNMSVIP). Finding x ∈ X such that 0∈F A x Mx, 2.1where A, F : X → X , η : X × X → X be single-valued mappings; M : X → 2X be anA, η -accretive set-valued mapping. A special case of problem 2.1 is the following: if X X ∗ is a Hilbert space, F 0 is the zero operator in X , and η x, y x − y, then problem 2.1 becomes the ...