Some New Results and Problems in the Theory of Differential-Delay Equations

Halanay, A.; Yorke, James A.

doi:10.1137/1013004

Cited by 66 publications

(23 citation statements)

References 29 publications

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“…The following equation was mentioned by Halanay and Yorke in [7]. Its analysis involves some nonstandard techniques.…”

Section: Let G Be a Closed Bounded Convex And Infinite Dimensional mentioning

confidence: 99%

Periodic solutions of autonomous functional differential equations

Nussbaum¹

1973

Bull. Amer. Math. Soc.

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ABSTRACT. The purpose of this note is to indicate some applications of a new fixed point theorem to the question of periodic solutions of nonlinear autonomous functional differential equations. The techniques developed give the standard periodicity examples in the literature and some new results, notably for the neutral case, which do not seem accessible by previous methods.1. If X is a Banach space and A is a bounded subset of X, define y (A), the measure of noncompactness of A 9 to be inf{d > 0:A has a finite covering by sets of diameter less than d). This is a notion due to C. Kuratowski ) y(cö(A)) = y(A) and (2) y(A + B) <. y (A) + y(B). It is trivially true that (3)y(AvB) = max{y(A%y(B)}.For applications it is sometimes convenient to generalize the above idea slightly. If ju is a function which assigns to each bounded subset A of X a real number fi(A\ we say that \x is a generalized measure of noncompactness if ii satisfies properties (1), (2)

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“…The following equation was mentioned by Halanay and Yorke in [7]. Its analysis involves some nonstandard techniques.…”

Section: Let G Be a Closed Bounded Convex And Infinite Dimensional mentioning

confidence: 99%

Periodic solutions of autonomous functional differential equations

Nussbaum¹

1973

Bull. Amer. Math. Soc.

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“…В ряде работ установлено, что для анализа динамики и свойств таких уравнений во многих случаях применимы методы теории уравнений с постоянным запаздыванием (см., например, [7][8][9][10]). Вопросы существования и устойчивости решений рассмат-ривались, например, в [5,[11][12][13]. Локальная динамика уравнения (1) при условии большого запаздывания изучена в [14].…”

Section: Introductionunclassified

Local bifurcations analysis of a state-dependent delay differential equation

Golubenets

2016

Aut. Control Comp. Sci.

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“…In fact, one of the great difficulties in studying them is the impossibility of describing a unified class sufficiently wide to be able to include a large amount of those equations which appear in the models. Some early works which deserve mention are [5] and [12]. The recent progress in the theoretical development of state-dependent delayed differential equations is highly developed in [7] with special emphasis on both the modelling and the emerging theory from the point of view of dynamical systems.…”

Section: Introductionmentioning

confidence: 99%

A threshold state-dependent delayed functional equation arising from marine population dynamics: modelling and analysis

2010

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This paper deals with a functional state-dependent delayed equation in which the delay is implicitly defined by a threshold condition. The equation originates in a class of age-structured population models in which the passage of individuals through the different stages of their life cycle occurs when some magnitude reaches a threshold value. The work analyses local existence of solutions, global existence of nonnegative solutions and uniqueness for smooth initial data. Some partial results on existence and stability of nonnegative stationary solutions are established through a linearized equation. This linearization is constructed from a differential formulation of the problem which is not equivalent to the original one, and the mathematical analysis is carried out in the setting of the C 0 -semigroup theory.

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Some New Results and Problems in the Theory of Differential-Delay Equations

Cited by 66 publications

References 29 publications

Periodic solutions of autonomous functional differential equations

Periodic solutions of autonomous functional differential equations

Local bifurcations analysis of a state-dependent delay differential equation

A threshold state-dependent delayed functional equation arising from marine population dynamics: modelling and analysis

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