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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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


Author: G. Edgar Parker
Journal: Proc. Amer. Math. Soc. 66 (1977), 33-37
MSC: Primary 47H99
MathSciNet review: 0482445
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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.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1977-0482445-4
PII: S 0002-9939(1977)0482445-4
Keywords: Nonlinear semigroup of transformations, Hille-Yosida theory, resolvent formula, trajectory, strongly continuous, exponential generator
Article copyright: © Copyright 1977 American Mathematical Society