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An example for a one-parameter nonexpansive semigroup. (English) Zbl 1106.47051

The author gives an example of a one-parameter nonexpansive semigroup \(\{ T(t): t\geq 0\}\) on a closed convex subset \(C\) of a Banach space such that \[ \displaystyle\lim\limits_{t\to\infty} \left\| \frac{1}{t}\int_0^t T(s)x\,ds-x\right\| =0 \] for some \(x\in C\), but \(x\) is not a common fixed point of the semigroup.

MSC:

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