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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 [S. Kamimura and W. Takahashi, SIAM J. Optim. 13, No. 3, 938–945 (2003; Zbl 1101.90083); F. Kohsaka and W. Takahashi, J. Nonlinear Convex Anal. 5, No. 3, 407–414 (2004; Zbl 1071.47062); S.-y. Matsushita and W. Takahashi, Approximation Theory 134, No. 2, 257–266 (2005; Zbl 1071.47063); K. Nakajo, J, Shimoji and W. Takahashi, Taiwanese J. Math. 10, No. 2, 339–360 (2006; Zbl 1109.47060)].

MSC:

49M15 Newton-type methods
47H05 Monotone operators and generalizations
90C25 Convex programming
47J25 Iterative procedures involving nonlinear operators
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