A class of one-parameter nonlinear semigroups with differentiable approximating semigroups

Abstract:

Suppose that T is a strongly continuous semigroup of transformations on a subset C of a Banach space X. For $\delta > 0$, consider ${U_\delta }(t) = \{ ({\delta ^{ - 1}}\smallint _0^\delta {g_x},{\delta ^{ - 1}}\smallint _0^\delta {g_{T(t)x}}):x \in C\}$ where $g_x$ denotes the trajectory of T from x. The class H of semigroups for which ${U_\delta }(t)$ is a function for $\delta > 0$ and $t \geqslant 0$ contains all strongly continuous linear semigroups and Webb’s nonlinear nonexpansive example with no dense set of differentiability. If $T \in H,{U_\delta } = \{ (t,{U_\delta }(t)):t \geqslant 0\}$ is a semigroup on $\{ {\delta ^{ - 1}}\smallint _0^\delta {g_x}:x \in C\}$ with continuously differentiable trajectories. Also, as $\{ {\delta _n}\} _{n = 1}^\infty$ converges to 0, the trajectories of $\{ {U_{{\delta _n}}}\} _{n = 1}^\infty$ uniformly approximate the trajectories of T.