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Strong convergence theorems for relatively nonexpansive mappings and Lipschitz-continuous monotone mappings in Banach spaces

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  • Ying Liu

    (Hebei University)

  • Hang Kong

    (Hebei University)

Abstract

In this paper, we introduce an iterative process for finding a common element of the set of fixed points of a relatively nonexpansive mapping and the set of solutions of the variational inequality for a Lipschitz-continuous, monotone mapping in a Banach space. We obtain a strong convergence theorem for three sequences generated by this process. Our results improve and extend the corresponding results announced by many others. A simple numerical example is given to support our theoretical results.

Suggested Citation

  • Ying Liu & Hang Kong, 2019. "Strong convergence theorems for relatively nonexpansive mappings and Lipschitz-continuous monotone mappings in Banach spaces," Indian Journal of Pure and Applied Mathematics, Springer, vol. 50(4), pages 1049-1065, December.
    • Handle: RePEc:spr:indpam:v:50:y:2019:i:4:d:10.1007_s13226-019-0373-0
    DOI: 10.1007/s13226-019-0373-0
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    References listed on IDEAS

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    1. Rapeepan Kraikaew & Satit Saejung, 2014. "Strong Convergence of the Halpern Subgradient Extragradient Method for Solving Variational Inequalities in Hilbert Spaces," Journal of Optimization Theory and Applications, Springer, vol. 163(2), pages 399-412, November.
    2. Y. Censor & A. Gibali & S. Reich, 2011. "The Subgradient Extragradient Method for Solving Variational Inequalities in Hilbert Space," Journal of Optimization Theory and Applications, Springer, vol. 148(2), pages 318-335, February.
    3. H. Iiduka, 2009. "Hybrid Conjugate Gradient Method for a Convex Optimization Problem over the Fixed-Point Set of a Nonexpansive Mapping," Journal of Optimization Theory and Applications, Springer, vol. 140(3), pages 463-475, March.
    4. Nakajo, Kazuhide, 2015. "Strong convergence for gradient projection method and relatively nonexpansive mappings in Banach spaces," Applied Mathematics and Computation, Elsevier, vol. 271(C), pages 251-258.
    5. W. Takahashi & M. Toyoda, 2003. "Weak Convergence Theorems for Nonexpansive Mappings and Monotone Mappings," Journal of Optimization Theory and Applications, Springer, vol. 118(2), pages 417-428, August.
    6. Lu-Chuan Ceng & Nicolas Hadjisavvas & Ngai-Ching Wong, 2010. "Strong convergence theorem by a hybrid extragradient-like approximation method for variational inequalities and fixed point problems," Journal of Global Optimization, Springer, vol. 46(4), pages 635-646, April.
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