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Strong convergence theorems by the hybrid method for families of mappings in Banach spaces. (English) Zbl 1221.47119
Summary: Let C be a nonempty closed convex subset of a uniformly convex Banach space E whose norm is Gâteaux differentiable 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 in mathematical programming, and present conditions on {T n } under which {x n } converges strongly to a common fixed point of {T n }, generalizing the results of [K. Nakajo, K. Shimoji and W. Takahashi, Taiwanese J. Math. 10, No. 2, 339–360 (2006; Zbl 1109.47060); S. Ohsawa and W. Takahashi, Arch. Math. 81, No. 4, 439–445 (2003; Zbl 1067.47080)].
MSC:
47J25Iterative procedures (nonlinear operator equations)
47H05Monotone operators (with respect to duality) and generalizations
49M05Numerical methods in calculus of variations based on necessary conditions