An example for a one-parameter nonexpansive semigroup (Q2494998)

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.