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Viscosity approximation of common fixed points of nonexpansive semigroups in Banach space. (English) Zbl 1161.47049
Taking as starting point the following viscosity implicit Mann-type iteration process $$ x_n=\alpha_n u+(1-\alpha_n) T(t_n)(x_n),\,n\geq 1, $$ introduced by {\it T. Suzuki} [Proc. Am. Math. Soc. 131, 2133--2136 (2002; Zbl 1031.47038)] for a nonexpansive semigroup $\{T(t):t\in \mathbb{R_{+}}\}$, the present authors consider two viscosity iteration processes, the first one an implicit iteration process $$ x_n=\alpha_n f(x_n)+(1-\alpha_n) T(t_n)(x_n),\,n\geq 1, $$ while the second one is an explicit iteration process $$ y_{n+1}=\alpha_n f(y_n)+(1-\alpha_n) T(t_n)(y_n),\,n\geq 1, $$ where $f$ is a contraction mapping. The main results of the paper are convergence theorems for these iterative processes toward a common fixed point of the semigroup which is also the unique solution of a certain variational inequality.

47J25Iterative procedures (nonlinear operator equations)
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
47H09Mappings defined by “shrinking” properties
47H20Semigroups of nonlinear operators
Full Text: DOI
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