## Viscosity approximation methods for a family of finite nonexpansive mappings in Banach spaces.(English)Zbl 1101.47053

Let $$E$$ be a reflexive Banach space with a uniformly Gâteaux differentiable norm, $$C$$ a closed convex subset of $$E$$, $$f: C \to C$$ a contraction, and $$T_1, \dots, T_N$$ nonexpansive mappings from $$C$$ into $$C$$. Suppose that the set $$F$$ of common fixed points of $$T_1, \dots, T_N$$ is nonempty. The authors study the iterative process $$x_{n+1}=\lambda_{n+1} f(x_n)+ (1-\lambda_{n+1}) T_{n+1} x_n$$, where $$x_0 \in C$$, $$\{ \lambda_n \} \in (0,1)$$ and $$T_n:= T_{n \text{ {mod}} N}$$. Under appropriate conditions on $$\{ \lambda_n \}$$, they prove that $$\{ x_n\}$$ converges strongly to an element of $$F$$.

### MSC:

 47J25 Iterative procedures involving nonlinear operators 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc. 47H10 Fixed-point theorems 49M05 Numerical methods based on necessary conditions 47J20 Variational and other types of inequalities involving nonlinear operators (general)
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