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Common fixed points of one-parameter nonexpansive semigroups in strictly convex Banach spaces. (English) Zbl 1130.47036

Summary: One of our main results is the following convergence theorem for one-parameter nonexpansive semigroups: let \(C\) be a bounded closed convex subset of a Hilbert space \(E\), and let \(\{T(t): t \in \mathbb{R}_+ \}\) be a strongly continuous semigroup of nonexpansive mappings on \(C\). Fix \(u \in C\) and \(t_1, t_2 \in \mathbb{R}_+\) with \(t_1 < t_2\). Define a sequence \(\{ x_n \}\) in \(C\) by \(x_{n} =(1-\alpha_n)/(t_2-t_1)\int_{t_1}^{t_2}T(s) x_n \,ds + \alpha_n u\) for \(n \in \mathbb{N}\), where \(\{ \alpha_n \}\) is a sequence in \((0,1)\) converging to \(0\). Then \(\{ x_n \}\) converges strongly to a common fixed point of \(\{T(t): t \in \mathbb{R}_+ \}\).

MSC:

47H20 Semigroups of nonlinear operators
47H10 Fixed-point theorems
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
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References:

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