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Strong Convergence of Iterative Algorithms for Variational Inequalities in Banach Spaces

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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.

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Authors and Affiliations

  1. Scientific Computing Key Laboratory of Shanghai Universities, and Department of Mathematics, Shanghai Normal University, Shanghai, 200234, China

    L. C. Ceng

  2. Department of Applied Mathematics, Chung Yuan Christian University, Chung-Li, Taiwan

    S. Schaible

  3. Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung, Taiwan

    J. C. Yao

Authors
  1. L. C. Ceng
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  2. S. Schaible
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  3. J. C. Yao
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Corresponding author

Correspondence to J. C. Yao.

Additional information

L.C. Ceng was partially supported by the National Science Foundation of China (10771141), PhD Program Foundation of Ministry of Education of China (20070270004), and Science and Technology Commission of Shanghai Municipality Grant (075105118). J.C. Yao was partially supported by Grant NSC 96-2628-E-110-014-MY3.

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Ceng, L.C., Schaible, S. & Yao, J.C. Strong Convergence of Iterative Algorithms for Variational Inequalities in Banach Spaces. J Optim Theory Appl 141, 265–283 (2009). https://doi.org/10.1007/s10957-008-9506-z

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