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Strong convergence of modified Mann iterations. (English) Zbl 1091.47055
Let $$X$$ be a real Banach space with a norm $$\|\cdot\|$$ and let $$C$$ be a nonempty, closed and convex subset of $$X$$. A mapping $$T:C\to C$$ is nonexpansive provided that $$\| Tx- Ty\|\leq\| x-y\|$$ for all $$x,y\in C$$. Assume that $$T$$ has at least one fixed point in $$C$$. The authors consider the following iteration sequence $$\{x_n\}$$ for $$T: x_0= x\in C$$, $$y_n= \alpha_nx_n+ (1-\alpha_n)Tx_n$$, $$x_{n+1}= \beta_n u+(1- \beta_n)y_n$$, where $$u$$ is an arbitrary fixed element in $$C$$ and $$\{\alpha_n\}$$, $$\{\beta_n\}$$ are two sequences in the interval $$(0,1)$$ converging to $$0$$ and such that $$\sum\alpha_n= \sum \beta_n= \infty$$. Moreover, $$\sum|\alpha_{n+1}- \alpha_n|< \infty$$, $$\sum|\beta_{n+1}- \beta_n|< \infty$$. Under the assumption that $$X$$ is uniformly smooth, it is shown that the sequence $$\{x_n\}$$ converges strongly to a fixed point of $$T$$. An analogous result is proved for accretive operators.

##### MSC:
 47J25 Iterative procedures involving nonlinear operators 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc. 46B25 Classical Banach spaces in the general theory 47H06 Nonlinear accretive operators, dissipative operators, etc.
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