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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.

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