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Strong convergence for gradient projection method and relatively nonexpansive mappings in Banach spaces

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  • Nakajo, Kazuhide

Abstract

Let C be a nonempty closed convex subset of a 2-uniformly convex and uniformly smooth Banach space E, A be a single valued monotone and Lipschitz continuous mapping of C into E* and T be a single valued relatively nonexpansive mapping of C into itself. In this paper, we consider the composition and the convex combination of T and the gradient projection method for A which Goldstein (1964) proposed and proved the strong convergence to a common element of solutions of the variational inequality problem for A and fixed points of T by the hybrid method in mathematical programming (Haugazeau, 1968). And we get several results which improve the well-known results in a 2-uniformly convex and uniformly smooth Banach space and a Hilbert space.

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  • 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.
    • Handle: RePEc:eee:apmaco:v:271:y:2015:i:c:p:251-258
    DOI: 10.1016/j.amc.2015.08.096
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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. 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.
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    Cited by:

    1. 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.

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