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An Extragradient Method and Proximal Point Algorithm for Inverse Strongly Monotone Operators and Maximal Monotone Operators in Banach Spaces

Abstract

We introduce an iterative scheme for finding a common element of the solution set of a maximal monotone operator and the solution set of the variational inequality problem for an inverse strongly-monotone operator in a uniformly smooth and uniformly convex Banach space, and then we prove weak and strong convergence theorems by using the notion of generalized projection. The result presented in this paper extend and improve the corresponding results of Kamimura et al. (2004), and Iiduka and Takahashi (2008). Finally, we apply our convergence theorem to the convex minimization problem, the problem of finding a zero point of a maximal monotone operator and the complementary problem.

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Correspondence to Somyot Plubtieng.

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Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Plubtieng, S., Sriprad, W. An Extragradient Method and Proximal Point Algorithm for Inverse Strongly Monotone Operators and Maximal Monotone Operators in Banach Spaces. Fixed Point Theory Appl 2009, 591874 (2009). https://doi.org/10.1155/2009/591874

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  • DOI: https://doi.org/10.1155/2009/591874

Keywords

  • Banach Space
  • Variational Inequality
  • Convergence Theorem
  • Strong Convergence
  • Monotone Operator