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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 =α n u+(1-α n )T(t n )(x n ),n1,

introduced by T. Suzuki [Proc. Am. Math. Soc. 131, 2133–2136 (2002; Zbl 1031.47038)] for a nonexpansive semigroup {T(t):t + }, the present authors consider two viscosity iteration processes, the first one an implicit iteration process

x n =α n f(x n )+(1-α n )T(t n )(x n ),n1,

while the second one is an explicit iteration process

y n+1 =α n f(y n )+(1-α n )T(t n )(y n ),n1,

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