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

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 $$u$$.

MSC:

 47H20 Semigroups of nonlinear operators 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc. 47J25 Iterative procedures involving nonlinear operators
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References:

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