Equations with unbounded delay: a survey

Corduneanu, C.; Lakshmikantham, V.

doi:10.1016/0362-546x(80)90001-2

Cited by 149 publications

(60 citation statements)

References 126 publications

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“…Thus, both of Theorems 2.1 and 2.5 contain the existence results for this global case; cf. [11,12,25,28,31,32,37,38], [50, Chapters B-IV.3, C-IV.3], [58,59,76,77,79,80,81,82], as well as [14,65,87] We note in passing that, for B ω−m-accretive, local existence results for (PFDE) can as well be achieved for F (globally defined and) Lipschitz on bounded sets by simply considering suitable truncations of F (being globally Lipschitz). 2.…”

Section: All Of Propositions (A)-(c) Of Theorem 21 Holdmentioning

confidence: 99%

Flow invariance for nonlinear partial differential delay equations

Ruess

2009

Trans. Amer. Math. Soc.

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Abstract. Several fundamental results on existence, flow-invariance, regularity, and linearized stability of solutions to the nonlinear partial differential delay equationu(t) + Bu(t) F (u t ), t ≥ 0, u 0 = ϕ, with B ⊂ X × X ω−accretive, are developed for a general Banach space X. In contrast to existing results, with the history-response F globally defined and, at least, Lipschitz on bounded sets, the results are tailored for situations with F defined on (possibly) thin subsets of the initial-history space E only, and are applied to place several classes of population models in their natural L 1 −setting. The main result solves the open problem of a subtangential condition for flow-invariance of solutions in the fully nonlinear case, paralleling those known for the cases of (a) no delay, (b) ordinary delay equations with B ≡ 0, and (c) the semilinear case.

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Section: All Of Propositions (A)-(c) Of Theorem 21 Holdmentioning

confidence: 99%

Flow invariance for nonlinear partial differential delay equations

Ruess

2009

Trans. Amer. Math. Soc.

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“…For the general correspondence between translation semigroups and first derivatives we refer also to [33 and 30]. Special translation semigroups on spaces of continuous functions were for the first time intensely studied by J. K. Hale [23] and afterwards by several others (see for example [14,39,7,11,8] and the the references therein). The special aspects of positivity are discussed in [17, 25, 26 and 27] (see also [32,).…”

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confidence: 99%

Translation Semigroups and Their Linearizations on Spaces of Integrable Functions

Grabosch¹

1989

Transactions of the American Mathematical Society

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ABSTRACT. Of concern is the unbounded operator A/ = f' with nonlinear domain D(A(f)} which is considered on the Banach space E of Bochner integrable functions on an interval with values in a Banach space F. Under the assumption that cJ> is a Lipschitz continuous operator from E to F, it is shown that A generates a strongly continuous translation semigroup (T(t))t>o. For linear operators cJ> several properties such as essential-compactness, positivity, and irreducibility of the semigroup (T(t))t>o depending on the operator cJ> are studied. It is shown that if F is a Banach lattice with order continuous norm, then (TI1 (t))t~O is the modulus semigroup of (T(t))t>o. Finally spectral properties of A are studied and the spectral bound s(A is linear and to a local stability result in the case where cJ> is Frechet differentiable.O. INTRODUCTION One of the basic examples in the theory of one-parameter semigroups on function spaces are semigroups which are generated by the first derivative on a suitable domain.At the beginning we want to review a particular simple example from the theory of strongly continuous semigroups which eventually will lead us to introduce a general notion of translation semigroups. We consider the unbounded opera-It is a straightforward computation to verify that Ao is the generator of a strongly continuous semigroup (To(t))t>o of linear operators explicitly given by the formula -[To(t)f](x) = { ~(X + t)

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“…For the basic theory of Volterra integral and Volterra integrodi¤erential equations, the reader is referred to the books by Burton [2,3], Corduneanu [4], Gripenberg, Londen and Sta¤ans [13], and Miller [23]. Also, for integrodi¤erential equations with unbounded delay, we choose to refer to the survey article by Corduneanu and Lakshmikantham [5] and the book by Lakshmikantham, Wen and Zhang [22].…”

Section: Introductionmentioning

confidence: 99%

“…Neutral integrodi¤erential equations with unbounded delay have appeared in several investigations; we choose to refer to the paper by Gopalsamy [10], the survey article by Corduneanu and Lakshmikantham [5], and the book by Lakshmikantham, Wen and Zhang [22].…”

Section: Introductionmentioning

confidence: 99%

Equations with unbounded delay: a survey

Cited by 149 publications

References 126 publications

Flow invariance for nonlinear partial differential delay equations

Flow invariance for nonlinear partial differential delay equations

Translation Semigroups and Their Linearizations on Spaces of Integrable Functions

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