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Strong convergence to common fixed points of infinite nonexpansive mappings and applications. (English) Zbl 0993.47037
This article deals with iterations \(x_{n+1}= \beta_n x+(1- \beta_n) W_nx_n\) \((n= 0,1,\dots)\), where \(W_n\) \((n=1,2,\dots)\) are mappings generated by the scheme \[ W_n= U_{n,1},\quad U_{n,k}= \alpha_k T_k U_{n,k+1}+(1- \alpha_k)I\quad (k= 1,\dots, n),\quad U_{n,n+1}= I, \] \(T_1,T_2,\dots\) are nonexpansive mappings of a convex subset of a Banach space \(E\) into itself, \(\bigcap^\infty_{n=1} \text{Fix }T_n\neq \emptyset\), \(\alpha_n\) satisfy the condition \(0< \alpha_n\leq b< 1\), \(\beta_n\) satisfy the conditions \(0\leq \beta_n\leq 1\), \(\lim_{n\to\infty} \beta_n= 0\), \(\sum^\infty_{n=1} |\beta_{n+1}- \beta_n|< \infty\), \(\sum^\infty_{n=1} \beta_n= \infty\). The basic result is the convergence of \(x_n\) to \(Px\), where \(P\) is the unique sunny nonexpansive retraction from \(C\) onto \(\bigcap^\infty_{n=1} \text{Fix }T_n\); it is assumed that the norm in \(E\) is uniformly convex and uniformly Gâteaux differentiable.

MSC:
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H10 Fixed-point theorems
47J25 Iterative procedures involving nonlinear operators
46B04 Isometric theory of Banach spaces
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