On multivalued nonlinear variational inclusion problems with -accretive mappings in Banach spaces
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Based on the notion of (A,η)-accretive mappings and the resolvent operators associated with (A,η)-accretive mappings due to Lan et al., we study a new class of multivalued nonlinear variational inclusion problems with (A,η)-accretive mappings in Banach spaces and construct some new iterative algorithms to approximate the solutions of the nonlinear variational inclusion problems involving (A,η)-accretive mappings. We also prove the existence of solutions and the convergence of the sequences generated by the algorithms in q-uniformly smooth Banach spaces. Copyright © 2006 Heng-You Lan. 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. 1. Introduction Recently, in order to study extensively variational inequalities and variational inclusions, which are providing mathematical models to some problems arising in economics, mechanics, and engineering science, Ding [1], Huang and Fang [10], Fang and Huang [3], Verma [14, 15], Fang and Huang [4, 5], Huang and Fang [9], Fang et al. [2] have introduced the concepts of η-subdifferential operators, maximal η-monotone operators, generalized monotone operators (named H-monotone operators), A-monotone operators, (H,η)-monotone operators in Hilbert spaces, H-accretive operators, generalized maccretive mappings and (H,η)-accretive operators in Banach spaces, and their resolvent operators, respectively. Very recently, Fang et al. [7], studied the (H,η)-monotone operators in Hilbert spaces, which are a special case of (H,η)-accretive operator [2]. Some works are motivated by this work and some related works. The iterative algorithms for the variational inclusions with H-accretive operators can be found in the paper [6]. Further, Lan et al. [11] introduced a new concept of (A,η)-accretive mappings, which generalizes the existing monotone or accretive operators, studied some properties of (A,η)-accretive mappings, and defined resolvent operators associated with (A,η)-accretive mappings. Moreover, by using the resolvent operator technique, many authors constructed some Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2006, Article ID 59836, Pages 1–12 DOI 10.1155/JIA/2006/59836
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On multivalued nonlinear variational inclusion problems
perturbed iterative algorithms for some nonlinear variational inclusions in Hilbert space or Banach spaces. For more detail, see, for example, [1–8, 10, 11, 14, 15] and the references therein. On the other hand, Lan et al. [12] introduced and studied some new iterative algorithms for solving a class of nonlinear variational inequalities with multivalued mappings in Hilbert spaces, and gave some convergence analysis of iterative sequences generated by the algorithms. Motivated and inspired by the above works, the purpose of this paper is to introduce the notion of (A,η)-accretive mappings and the resolvent operators associated with (A,η)-accretive mappings due to L
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