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

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.