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 \(F\).


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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