Strong Convergence of Iterative Algorithms for Variational Inequalities in Banach Spaces

L. C. Ceng (), S. Schaible and J. C. Yao ()
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L. C. Ceng: Shanghai Normal University
S. Schaible: Chung Yuan Christian University
J. C. Yao: National Sun Yat-Sen University

Journal of Optimization Theory and Applications, 2009, vol. 141, issue 2, No 3, 265-283

Abstract: Abstract Let C be a nonempty closed convex subset of a Banach space E with the dual E *, let T:C→E * be a Lipschitz continuous mapping and let S:C→C be a relatively nonexpansive mapping. In this paper, by employing the notion of generalized projection operator, we study the following variational inequality (for short, VI(T−f,C)): find x∈C such that $$\langle y-x,Tx-f\rangle\geq0,\quad\mbox{for all }y\in C,$$ where f∈E * is a given element. Utilizing the modified Ishikawa iteration and the modified Halpern iteration for relatively nonexpansive mappings, we propose two modified versions of J.L. Li’s (J. Math. Anal. Appl. 295:115–126, 2004) iterative algorithm for finding approximate solutions of VI(T−f,C). Moreover, it is proven that these iterative algorithms converge strongly to the same solution of VI(T−f,C), which is also a fixed point of S.

Keywords: Generalized projection operators; Iterative algorithms; Variational inequalities; Relatively nonexpansive mappings; Banach spaces; Strong convergence (search for similar items in EconPapers)
Date: 2009
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DOI: 10.1007/s10957-008-9506-z

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