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Journal of Convex Analysis 26 (2019), No. 1, 201--216
Copyright Heldermann Verlag 2019

Strong Convergence Theorems by Hybrid Methods for New Demimetric Mappings in Banach Spaces

Wataru Takahashi
Center for Fundamental Science, Research Center for Nonlinear Analysis and Optimization, Kaohsiung Medical University, Kaohsiung 80708, Taiwan
and: Dept. of Mathematical and Computing Sciences, Tokyo Institute of Technology, Ookayama, Meguro-ku, Tokyo 152-8552, Japan

Using a new nonlinear mapping called generalized demimetric and the C-Q method, we first prove a strong convergence theorem for finding a fixed point for the mapping in a Banach space which generalizes simultaneously the results by Nakajo and Takahashi [Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. Math. Anal. Appl. 279 (2003) 372--379], and Solodov and Svaiter [Forcing strong convergence of proximal point iterations in a Hilbert space, Math. Programming Ser. A 87 (2000) 189--202] in a Hilbert space. Furthermore, using the mapping and the shrinking projection method, we prove another strong convergence theorem in a Banach space. We apply these results to obtain new strong convergence theorems in a Hilbert space and a Banach space.

Keywords: Fixed point, demimetric mapping, maximal monotone operator, metric resolvent, metric projection, hybrid method, shrinking projection method, duality mapping.

MSC: 47H05, 47H10

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