Oscillation and Global Attractivity in Stage-Structured Population Models

AbstractStage-structured models of population growth have been considered in the constant delay and state-dependent delay cases, when modeled by retarded functional differential equations. In the first case we settle a conjecture posed by Aiello and Freedman [1] by showing the existence of oscillatory solutions. In the second case, we show that under suitable criteria, all positive solutions tend to a global attractor.

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A Note on Existence of Mild Solutions for Second-Order Neutral Integro-Differential Evolution Equations with State-Dependent Delay

Fractal and Fractional ◽

10.3390/fractalfract5030126 ◽

2021 ◽

Vol 5 (3) ◽

pp. 126

Author(s):

Shahram Rezapour ◽

Hernán R. Henríquez ◽

Velusamy Vijayakumar ◽

Kottakkaran Sooppy Nisar ◽

Anurag Shukla

Keyword(s):

Differential Evolution ◽

Evolution Equations ◽

Mild Solutions ◽

Functional Differential Equations ◽

Second Order ◽

State Dependent ◽

Retarded Functional Differential Equations ◽

State Dependent Delay ◽

Functional Differential ◽

Infinite State

This article is mainly devoted to the study of the existence of solutions for second-order abstract non-autonomous integro-differential evolution equations with infinite state-dependent delay. In the first part, we are concerned with second-order abstract non-autonomous integro-differential retarded functional differential equations with infinite state-dependent delay. In the second part, we extend our results to study the second-order abstract neutral integro-differential evolution equations with state-dependent delay. Our results are established using properties of the resolvent operator corresponding to the second-order abstract non-autonomous integro-differential equation and fixed point theorems. Finally, an application is presented to illustrate the theory obtained.

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On fractional differential equations with state-dependent delay via Kuratowski measure of noncompactness

Filomat ◽

10.2298/fil1702451b ◽

2017 ◽

Vol 31 (2) ◽

pp. 451-460 ◽

Cited By ~ 1

Author(s):

Mohammed Belmekki ◽

Kheira Mekhalfi

Keyword(s):

Differential Equations ◽

Measure Of Noncompactness ◽

Mild Solutions ◽

Functional Differential Equations ◽

Kuratowski Measure Of Noncompactness ◽

State Dependent ◽

State Dependent Delay ◽

Liouville Fractional Derivative ◽

Fractional Differential ◽

Functional Differential

This paper is devoted to study the existence of mild solutions for semilinear functional differential equations with state-dependent delay involving the Riemann-Liouville fractional derivative in a Banach space and resolvent operator. The arguments are based upon M?nch?s fixed point theoremand the technique of measure of noncompactness.

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Periodic Solutions for a Class of Functional Differential Equations with State-Dependent Delay Close to Zero

Mathematical Models and Methods in Applied Sciences ◽

10.1142/s0218202503002738 ◽

2003 ◽

Vol 13 (06) ◽

pp. 807-841 ◽

Cited By ~ 6

Author(s):

R. Ouifki ◽

M. L. Hbid

Keyword(s):

Differential Equation ◽

Hopf Bifurcation ◽

Periodic Solutions ◽

Functional Differential Equations ◽

Delay Equation ◽

Constant Delay ◽

State Dependent ◽

State Dependent Delay ◽

Existence Of Periodic Solutions ◽

Functional Differential

The purpose of the paper is to prove the existence of periodic solutions for a functional differential equation with state-dependent delay, of the type [Formula: see text] Transforming this equation into a perturbed constant delay equation and using the Hopf bifurcation result and the Poincaré procedure for this last equation, we prove the existence of a branch of periodic solutions for the state-dependent delay equation, bifurcating from r ≡ 0.

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Perturbation theory for dual semigroups II. Time-dependent perturbations in the sun-reflexive case

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500026731 ◽

1988 ◽

Vol 109 (1-2) ◽

pp. 145-172 ◽

Cited By ~ 23

Author(s):

Ph. Clément ◽

O. Diekmann ◽

M. Gyllenberg ◽

H. J. A. M. Heijmans ◽

H. R. Thieme

Keyword(s):

Banach Space ◽

Functional Differential Equations ◽

Time Dependent ◽

Evolutionary System ◽

Strongly Continuous Semigroups ◽

Structured Population ◽

Variation Of Constants ◽

Retarded Functional Differential Equations ◽

Functional Differential ◽

Age Structured

SynopsisWe consider time-dependent perturbations of generators of strongly continuous semigroups on a Banach space. The perturbations map the Banach space into a bigger space, which is the second dual of the original space in a specific semigroup sense. Using the theory of dual semigroups we show that the solutions of a generalised variation-of-constants formuladefine an evolutionary system. We investigate continuity and differentiability propertiesof this evolutionary system and its dual system and examine in what sense the perturbed generator and its adjoint generate these evolutionary systems. It is shown that the results apply naturally to retarded functional differential equations and age structured population dynamics.

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