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

Author

Listed:
  • 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)

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.

Suggested Citation

  • L. C. Ceng & S. Schaible & J. C. Yao, 2009. "Approximate Solutions of Variational Inequalities on Sets of Common Fixed Points of a One-Parameter Semigroup of Nonexpansive Mappings," Journal of Optimization Theory and Applications, Springer, vol. 143(2), pages 245-263, November.
    • Handle: RePEc:spr:joptap:v:143:y:2009:i:2:d:10.1007_s10957-009-9581-9
    DOI: 10.1007/s10957-009-9581-9
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