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Strong Convergence of Iterative Algorithms for Variational Inequalities in Banach Spaces

Author

Listed:
  • L. C. Ceng

    (Shanghai Normal University)

  • S. Schaible

    (Chung Yuan Christian University)

  • J. C. Yao

    (National Sun Yat-Sen University)

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.

Suggested Citation

  • L. C. Ceng & S. Schaible & J. C. Yao, 2009. "Strong Convergence of Iterative Algorithms for Variational Inequalities in Banach Spaces," Journal of Optimization Theory and Applications, Springer, vol. 141(2), pages 265-283, May.
    • Handle: RePEc:spr:joptap:v:141:y:2009:i:2:d:10.1007_s10957-008-9506-z
    DOI: 10.1007/s10957-008-9506-z
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    References listed on IDEAS

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    1. L. C. Zeng & J. C. Yao, 2007. "Existence Theorems for Variational Inequalities in Banach Spaces," Journal of Optimization Theory and Applications, Springer, vol. 132(2), pages 321-337, February.
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