## 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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