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Hybrid Extragradient Iterative Algorithms for Variational Inequalities, Variational Inclusions, and Fixed‐Point Problems

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  • Lu-Chuan Ceng
  • Ching-Feng Wen

Abstract

We investigate the problem of finding a common solution of a general system of variational inequalities, a variational inclusion, and a fixed‐point problem of a strictly pseudocontractive mapping in a real Hilbert space. Motivated by Nadezhkina and Takahashi′s hybrid‐extragradient method, we propose and analyze new hybrid‐extragradient iterative algorithm for finding a common solution. It is proven that three sequences generated by this algorithm converge strongly to the same common solution under very mild conditions. Based on this result, we also construct an iterative algorithm for finding a common fixed point of three mappings, such that one of these mappings is nonexpansive, and the other two mappings are strictly pseudocontractive mappings.

Suggested Citation

  • Lu-Chuan Ceng & Ching-Feng Wen, 2012. "Hybrid Extragradient Iterative Algorithms for Variational Inequalities, Variational Inclusions, and Fixed‐Point Problems," Journal of Applied Mathematics, John Wiley & Sons, vol. 2012(1).
    • Handle: RePEc:wly:jnljam:v:2012:y:2012:i:1:n:924309
    DOI: 10.1155/2012/924309
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    References listed on IDEAS

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    1. Heinz H. Bauschke & Patrick L. Combettes, 2001. "A Weak-to-Strong Convergence Principle for Fejér-Monotone Methods in Hilbert Spaces," Mathematics of Operations Research, INFORMS, vol. 26(2), pages 248-264, May.
    2. M. V. Solodov & B. F. Svaiter, 2000. "An Inexact Hybrid Generalized Proximal Point Algorithm and Some New Results on the Theory of Bregman Functions," Mathematics of Operations Research, INFORMS, vol. 25(2), pages 214-230, May.
    3. N. Nadezhkina & W. Takahashi, 2006. "Weak Convergence Theorem by an Extragradient Method for Nonexpansive Mappings and Monotone Mappings," Journal of Optimization Theory and Applications, Springer, vol. 128(1), pages 191-201, January.
    4. Eberhard Zeidler, 1985. "Nonlinear Functional Analysis and its Applications," Springer Books, Springer, number 978-1-4612-5020-3, March.
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