NONLINEAR STRONG ERGODIC THEOREMS WITH COMPACT DOMAINS

Let $C $ be a nonempty closed convex subset of a real Banach space $E $. Then a mapping $T:Carrow C $ is called nonexpansive if $||Tx-^{\tau}y||\leq||x-y|| $ for all $x, $ $y\in C $. We denote by $F(T) $ the set of fixed points of $T$. For any $x\in C $ , the $\omega$-limit set of $x $ is defined by $\omega(x)= $ { $z \in C:z=\lim_{iarrow\infty}\tau^{n_{i}}X $ with $n_{i}arrow\infty $ as $iarrow\infty $}