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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Operator semigroups for functional-differential equations with delay

Authors: W. M. Ruess and W. H. Summers
Journal: Trans. Amer. Math. Soc. 341 (1994), 695-719
MSC: Primary 34K30; Secondary 47H20, 47N20
MathSciNet review: 1214785
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Abstract: We show that a strongly continuous operator semigroup can be associated with the functional differential delay equation (FDE)

$\displaystyle \left\{ {\begin{array}{*{20}{c}} {x\prime(t) + ax(t) + Bx(t) \bac... ...{\mathbb{R}^ - }}} = \varphi \in E} \hfill & {} \hfill \\ \end{array} } \right.$

under local conditions which give wide latitude to those subsets of the state space $ X$ and initial data space $ E$, respectively, where (a) the (generally multivalued) operator $ B \subseteq X \times X$ is defined and accretive, and (b) the historyresponsive function $ F:D(F) \subseteq E \to X$ is defined and Lipschitz continuous. The associated semigroup is then used to investigate existence and uniqueness of solutions to (FDE). By allowing the domain of the solution semigroup to be restricted according to specific local properties of $ B$ and $ F$, moreover, our methods automatically lead to assertions on flow invariance. We illustrate our results through applications to the Goodwin oscillator and a single species population model.

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Article copyright: © Copyright 1994 American Mathematical Society

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