Hassan A. El-Morshedy

Convergence to equilibria in discrete population models

For a family of difference equations where and is continuous and decreasing, we find sufficient conditions for the convergence of all solutions to the unique positive equilibrium. In particular, we improve, up to our knowledge, all previous results on the global asymptotic stability of the equilibrium in the particular cases of the discrete Mackey–Glass and Lasota–Wazewska models in blood-cells production.

Global attractors for difference equations dominated by one-dimensional maps

The global attractivity character of nonlinear higher order difference equations of the form is investigated when g is dominated by an interval scalar map. Some basic properties of the interval map are obtained and used to prove new global attractivity criteria for the above equation with no monotonicity restrictions on g. Our results are applied to many models from mathematical biology and economy. The derived global attractivity criteria of these models are either new or improve substantially known ones.

New oscillation criterion for delay differential equations with non-monotone arguments

Abstract We investigate the oscillation of a first order delay differential equation with non-negative coefficient and non-monotone arguments. New oscillation criterion of lim sup type is obtained. An example is given to show the applicability and strength of the obtained condition over known ones.

On the global attractivity and oscillations in a class of second-order difference equations from macroeconomics

New global attractivity criteria are obtained for the second-order difference equation via a Lyapunov-like method. Some of these results are sharp and support recent related conjectures. Also, a necessary and sufficient condition for the oscillation of this equation is obtained using comparison with a second-order linear difference equation with positive coefficients.

On the distance between adjacent zeros of solutions of first order differential equations with distributed delays

We estimate the distance between adjacent zeros of all solutions of the first order differential equation x′(t) + ∫ t h(t) x(s)dsR(t, s) = 0. This form makes it possible to study equations with both discrete and continuous distributions of the delays. The obtained results are new and improve several known estimations. Some illustrative examples are given to show the advantages of our results over the known ones.

The global attractivity of a rational difference equation with variable delays

In this paper, we study the rational difference equation where p, , and are sequences of positive integers such that , for all and . It is proved that this equation has either one or two equilibria. The unique equilibrium point is proved to be the global attractor of all solutions. The dynamics, when the system has two equilibria, is also investigated.