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On strong convergence to common fixed points of nonexpansive semigroups in Hilbert spaces. (English) Zbl 1031.47038

Let \(C\) be a closed convex subset of a Hilbert space \(H\). Also, let \(\{T(t);t\geq 0\}\) be a strongly continuous semigroup of nonexpansive maps on \(C\) such that \(F(T)\) (the set of its common fixed points) is nonvoid. Fix \(u\in C\) and define the sequence \((u_n)\) in \(C\) as \((u_n=(1-\alpha_n)T(t_n)u_n+\alpha_n u,\;n\in \mathbb{N})\), where \((\alpha_n)\) in \(]0,1[\) and \((t_n)\) in \(]0,\infty[\) satisfy \(\lim_n (t_n)=\lim_n(\alpha_n/t_n)=0\). Then, \((u_n)\) converges strongly to the element of \(F(T)\) nearest to \(u\).

MSC:

47H20 Semigroups of nonlinear operators
47H10 Fixed-point theorems
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References:

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