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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



A characterization of nonlinear semigroups with smoothing effect

Author: Michiaki Watanabe
Journal: Proc. Amer. Math. Soc. 103 (1988), 153-159
MSC: Primary 47H20; Secondary 47H06
MathSciNet review: 938661
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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

$\displaystyle x \in C,\;S(t)x \in V\;{\text{a}}{\text{.e}}{\text{.}}\;{\text{t > 0,}}$

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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Keywords: Nonlinear semigroup, accretive operator, smoothing effect
Article copyright: © Copyright 1988 American Mathematical Society