Electron. J. Diff. Equ., Vol. 2014 (2014), No. 177, pp. 1-13.

Global attractivity for nonlinear differential equations with a nonlocal term

Boumediene Abdellaoui, Tarik Mohamed Touaoula

Abstract:
In this article we analyze the dynamics of the problem
$$\displaylines{ x'(t)=-(\delta+\beta(x(t)))x(t)+\theta\int_{0}^{\tau}f(a)x(t-a)\beta(x(t-a))da, \quad t> \tau, \cr x(t)=\phi(t),\quad 0 \leq t\leq \tau, }$$
where $\delta,\theta$ are positive constants, and $\beta, \phi, f$ are positives continuous functions. The main results obtained in this paper are the following: (1) Using the Laplace transform, we prove the global asymptotic stability of the trivial steady state. (2) Under some additional hypotheses on the data and by constructing a Lyapunov functional, we show the asymptotic stability of the positive steady state. We conclude by applying our results to mathematical models of hematopoieses and Nicholson's blowflies.

Submitted June 26, 2014. Published August 15, 2014.
Math Subject Classifications: 34K20, 92C37.
Key Words: Global stability; Lyapunov function; asymptotic analysis; Laplace transform, Nicholson's blowflies model.

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Boumediene Abdellaoui
Laboratoire d'Analyse Nonlinéaire et Mathématiques Appliquées
Département de Mathématiques
Université Abou Bakr Belkaid
Tlemcen, Tlemcen 13000, Algeria
email: boumediene.abdellaoui@uam.es
Tarik Mohamed Touaoula
Laboratoire d'Analyse Nonlinéaire et Mathématiques Appliquées
Département de Mathématiques
Université Abou Bakr Belkaid
Tlemcen, Tlemcen 13000, Algeria
email: touaoula_tarik@yahoo.fr, tarik.touaoula@univ-tlemcen.dz

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