Asymptotic behavior of almost-orbits of nonlinear semigroups of non-Lipschitzian mappings in Hilbert spaces

Abstract:

Let $C$ be a nonempty closed convex subset of a Hilbert space $H$, $\mathcal {F} = \{ T(t):t \geqslant 0\}$ be a continuous nonlinear asymptotically nonexpansive semigroup acting on $C$ with a nonempty fixed point set $F(\mathcal {F})$, and $u:[0,\infty ) \to C$ be an almost-orbit of $\mathcal {F}$. Then $\{ u(t)\}$ almost converges weakly to a fixed point of $\mathcal {F}$, i.e., there exists an element $y$ in $F(\mathcal {F})$ such that \[ {\text {weak-}}\lim \frac {1} {t}\int _0^t {u(r + h)dr = y\quad {\text {uniformly for }}h \geqslant 0.} \] This implies that $\{ u(t)\}$ converges weakly to a fixed point of $\mathcal {F}$ if and only if $\{ u(t + h) - u(t)\}$ converges weakly to zero as $t$ tends to infinity for each $h \geqslant 0$.