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Strong convergence theorems obtained by a generalized projections hybrid method for families of mappings in Banach spaces. (English) Zbl 1201.49033
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:
 49M15 Newton-type methods in calculus of variations 47H05 Monotone operators (with respect to duality) and generalizations 90C25 Convex programming 47J25 Iterative procedures (nonlinear operator equations)
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##### References:
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