Approximate Solutions of Variational Inequalities on Sets of Common Fixed Points of a One-Parameter Semigroup of Nonexpansive Mappings

L. C. Ceng (), S. Schaible () and J. C. Yao ()
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L. C. Ceng: Shanghai Normal University, and Scientific Computing Key Laboratory of Shanghai Universities
S. Schaible: Chung Yuan Christian University
J. C. Yao: National Sun Yat-sen University

Journal of Optimization Theory and Applications, 2009, vol. 143, issue 2, No 2, 245-263

Abstract: Abstract Let $\mathcal{T}$ be a one-parameter semigroup of nonexpansive mappings on a nonempty closed convex subset C of a strictly convex and reflexive Banach space X. Suppose additionally that X has a uniformly Gâteaux differentiable norm, C has normal structure, and $\mathcal{T}$ has a common fixed point. Then it is proved that, under appropriate conditions on nonexpansive semigroups and iterative parameters, the approximate solutions obtained by the implicit and explicit viscosity iterative processes converge strongly to the same common fixed point of $\mathcal{T}$ , which is a solution of a certain variational inequality.

Keywords: Viscosity approximation methods; Fixed-point problems; Variational inequalities; Nonexpansive mappings; Strong convergence; Reflexive and strictly convex Banach spaces; Uniformly Gâteaux differentiable norms (search for similar items in EconPapers)
Date: 2009
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DOI: 10.1007/s10957-009-9581-9

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