A characterization of nonlinear semigroups with smoothing effect

Watanabe, Michiaki

doi:10.1090/S0002-9939-1988-0938661-5

A characterization of nonlinear semigroups with smoothing effect
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Proc. Amer. Math. Soc. 103 (1988), 153-159 Request permission

Abstract:

Let $\{ S(t):t > 0\}$ be a nonlinear semigroup of operators mapping a closed subset $C$ of a real Banach space $X$ into itself. Conditions are found for an accretive operator in $X$ to be the generator of $\{ S(t):t > 0\}$ with smoothing effect: For each \[ x \in C,\;S(t)x \in V\;{\text {a}}{\text {.e}}{\text {.}}\;{\text {t > 0,}}\] among other things, where $V$ is a Banach space imbedded continuously in $X$. The conditions contain a Gårding-type inequality, and are shown also to be necessary if $C$ is a closed convex subset of a "nice" Banach space $X$.

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