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Convergence to common fixed point of nonexpansive semigroups. (English) Zbl 1204.47076

Summary: Let E be a uniformly convex Banach space whose norm is uniformly Gâteaux differentiable, C be a closed convex subset of E, 𝒮={T(s):s0} be a nonexpansive semigroup on C such that the set of common fixed points of {T(s):s0} is nonempty. Let f:CC be a contraction, {α n }, {β n }, {t n } be real sequences such that 0<α n ,β n 1, lim n α n =0, lim n β n =0 and lim n t n =, y 0 C. In this paper, we show that the two iterative sequences defined as follows:

x n =α n f(x n )+(1-α n )1 t n 0 t n T(s)x n ds,y n-1 =β n f(y n )+(1-β n )1 t n 0 t n T(s)y n ds,

converge strongly to a common fixed point of {T(s):s0} which solves some variational inequality when {α n }, {β n } satisfy some appropriate conditions.

MSC:
47H20Semigroups of nonlinear operators
47J25Iterative procedures (nonlinear operator equations)
65J15Equations with nonlinear operators (numerical methods)
47H10Fixed point theorems for nonlinear operators on topological linear spaces
47H09Mappings defined by “shrinking” properties