## 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

### Citations:

Zbl 1101.90083; Zbl 1071.47062; Zbl 1071.47063; Zbl 1109.47060
Full Text:

### References:

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