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Weak convergence of a projection algorithm for variational inequalities in a Banach space. (English) Zbl 1129.49012

Summary: Let C be a nonempty, closed convex subset of a Banach space E. In this paper, motivated by Ya. I. Alber [Metric and generalized projection operators in Banach spaces: Properties and applications, in: A.G. Kartsatos (ed.), Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, in: Lect. Notes Pure Appl. Math. 178, Dekker, New York, 15–50 (1996; Zbl 0883.47083)], we introduce the following iterative scheme for finding a solution of the variational inequality problem for an inverse-strongly-monotone operator A in a Banach space: x 1 =xC and

x n+1 =Π C J -1 (Jx n -λ n Ax n ),

for every n=1,2,, where Π C is the generalized projection from E onto C, J is the duality mapping from E into E * and {λ n } is a sequence of positive real numbers. Then we show a weak convergence theorem (Theorem 3.1). Finally, using this result, we consider the convex minimization problem, the complementarity problem, and the problem of finding a point uE satisfying 0=Au.

MSC:
49J40Variational methods including variational inequalities
47J20Inequalities involving nonlinear operators
49M15Newton-type methods in calculus of variations
49J45Optimal control problems involving semicontinuity and convergence; relaxation
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