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Zbl 1201.49033
Matsushita, Shin-Ya; Nakajo, Kazuhide; Takahashi, Wataru
Strong convergence theorems obtained by a generalized projections hybrid method for families of mappings in Banach spaces.
(English)
[J] Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 73, No. 6, 1466-1480 (2010). ISSN 0362-546X

Summary: Let $C$ be a nonempty, closed and convex subset of a uniformly convex and smooth Banach space and let $\{T_n\}$ be a family of mappings of $C$ into itself such that the set of all common fixed points of $\{T_n\}$ is nonempty. We consider a sequence $\{x_n\}$ generated by the hybrid method by generalized projection in mathematical programming. We give conditions on $\{T_n\}$ under which $\{x_n\}$ converges strongly to a common fixed point of $\{T_n\}$ and generalize the results given in [{\it S. Kamimura} and {\it W. Takahashi}, SIAM J. Optim. 13, No.~3, 938--945 (2003; Zbl 1101.90083); {\it F. Kohsaka} and {\it W. Takahashi}, J. Nonlinear Convex Anal. 5, No.~3, 407--414 (2004; Zbl 1071.47062); {\it S.-y. Matsushita} and {\it W. Takahashi}, Approximation Theory 134, No.~2, 257--266 (2005; Zbl 1071.47063); {\it K. Nakajo, J, Shimoji} and {\it W. Takahashi}, Taiwanese J. Math. 10, No.~2, 339--360 (2006; Zbl 1109.47060)].
MSC 2000:
*49M15 Methods of Newton-Raphson, Galerkin and Ritz types
47H05 Monotone operators (with respect to duality)
90C25 Convex programming
47J25 Methods for solving nonlinear operator equations (general)

Keywords: strong convergence; hybrid method; generalized projection; monotone operators; proximal point algorithm; relatively nonexpansive mappings; $W$-mappings; convex feasibility problem

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