Browder’s convergence for uniformly asymptotically regular nonexpansive semigroups in Hilbert spaces. (English) Zbl 1206.47058

Acedo, Genaro López; Suzuki, Tomonari

Fixed Point Theory Appl. 2010, Article ID 418030, 8 p. (2010).

Let \(C\) be a closed convex subset of a Hilbert space and \((T(t):t\geq 0)\) be a uniformly asymptotically regular nonexpansive semigroup on \(C\) with \(F:=\cap\{F(T(t)):t\geq 0\}\neq \emptyset\). Let \((a_n)\subset (0,1)\) and \((t_n)\subset (0,\infty)\) be sequences with \(\lim_n(a_n)=\lim_n(a_n/t_n)=0\). Then, for each \(u\in C\), the implicit sequence
\[ x_n=a_nu+(1-a_n)T(t_n)x_n, \quad n\geq 0, \]
converges strongly to an element of \(F\), nearest to \(u\).

Reviewer: Mihai Turinici (Iaşi)

MSC:

47H20	Semigroups of nonlinear operators
47H09	Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47J25	Iterative procedures involving nonlinear operators

Keywords:

Hilbert space; closed convex set; one-parameter strongly continuous semigroup of nonexpansive mappings; common fixed point; implicit sequence; Opial property; strong convergence

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