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Proceedings of the American Mathematical Society

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Regularizing properties of nonlinear semigroups

Author: Semion Gutman
Journal: Proc. Amer. Math. Soc. 101 (1987), 51-56
MSC: Primary 47H20; Secondary 34G20, 35R20, 47H06
MathSciNet review: 897069
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Abstract: It is known that some classes of $ m$-accretive operators $ A$ generate Lipschitz continuous semigroups of contractions; that is $ \vert\vert S(t + h)x - S(t)x\vert\vert \leqslant L(\vert\vert x\vert\vert)h/t,0 \leqslant t \leqslant t + h \leqslant T,x \in \overline {D(A)} $. If the underlying Banach spaces $ X$ and $ {X^*}$ are uniformly convex and an $ m$-accretive operator $ B$ is bounded, we prove, in particular, that the semigroup generated by $ A + B$ is Hölder continuous. The proof is based on a result on the structure of accretive operators obtained via the Kuratowski-Ryll-Nardzewski Selection Theorem. Also, we consider some applications of these results to the existence of solutions of $ u' + Au + Bu \mathrel\backepsilon Cu,u(0) = {u_0}$.

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Keywords: Nonlinear semigroup, regularity, $ m$-accretive, evolution equation, bounded perturbation
Article copyright: © Copyright 1987 American Mathematical Society

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