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Zbl 1221.47119
Nakajo, Kazuhide; Shimoji, Kazuya; Takahashi, Wataru
Strong convergence theorems by the hybrid method for families of mappings in Banach spaces. (English)
[J] Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 71, No. 3-4, A, 812-818 (2009). ISSN 0362-546X

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 [{\it K. Nakajo}, {\it K. Shimoji} and {\it W. Takahashi}, Taiwanese J. Math. 10, No.~2, 339--360 (2006; Zbl 1109.47060); {\it S. Ohsawa} and {\it W. Takahashi}, Arch. Math. 81, No.~4, 439--445 (2003; Zbl 1067.47080)].

MSC 2000:: *47J25 Methods for solving nonlinear operator equations (general)
47H05 Monotone operators (with respect to duality)
49M05 Methods of successive approximation based on necessary conditions

Keywords: strong convergence; hybrid method; monotone operator; proximal point algorithm; convex feasibility problem

Citations: Zbl 1109.47060; Zbl 1067.47080

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