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Convergence on composite iterative schemes for nonexpansive mappings in Banach spaces. (English) Zbl 1203.47053
Let $$E$$ be a reflexive Banach space with a uniformly Gateaux differentiable norm. Suppose that every weakly compact convex subset of $$E$$ has the fixed point property for nonexpansive mappings. Let $$C$$ be a nonempty closed convex subset of $$E$$, $$f: C \to C$$ be a contractive mapping (or a weakly contractive mapping), and $$T: C \to C$$ be a nonexpansive mapping with nonempty fixed point set $$F(T)$$. Let the sequence $$\{x_n\}$$ be generated by the following composite iterative scheme: \left\{\begin{aligned} &y_n=\lambda_n f(x_n)+(1-\lambda_n)Tx_n,\\ &x_{n+1}=(1-\beta_n)y_n+\beta_n Ty_n,\end{aligned}\right.\quad n\geq0. It is proved that $$\{x_n\}$$ converges strongly to a point in $$F(T)$$, which is a solution of a certain variational inequality, provided that the sequence $$\{\lambda_n\}\subset(0,1)$$ satisfies $$\lim_{n\rightarrow\infty}\lambda_n=0$$ and $$\sum_{n=1}^\infty\lambda_n=\infty$$, $$\{\beta_n\}\subset[0,a)$$ for some $$0<a<1$$, and the sequence $$\{x_n\}$$ is asymptotically regular.

##### MSC:
 47J25 Iterative procedures involving nonlinear operators 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc.
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