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Averaging for ordinary differential equations perturbed by a small parameter

Mustapha Lakrib, Tahar Kherraz, Amel Bourada (2016)

Mathematica Bohemica

In this paper, we prove and discuss averaging results for ordinary differential equations perturbed by a small parameter. The conditions we assume on the right-hand sides of the equations under which our averaging results are stated are more general than those considered in the literature. Indeed, often it is assumed that the right-hand sides of the equations are uniformly bounded and a Lipschitz condition is imposed on them. Sometimes this last condition is relaxed to the uniform continuity in...

Behaviour of solutions of linear differential equations with delay

Josef Diblík (1998)

Archivum Mathematicum

This contribution is devoted to the problem of asymptotic behaviour of solutions of scalar linear differential equation with variable bounded delay of the form x ˙ ( t ) = - c ( t ) x ( t - τ ( t ) ) ( * ) with positive function c ( t ) . Results concerning the structure of its solutions are obtained with the aid of properties of solutions of auxiliary homogeneous equation y ˙ ( t ) = β ( t ) [ y ( t ) - y ( t - τ ( t ) ) ] where the function β ( t ) is positive. A result concerning the behaviour of solutions of Eq. (*) in critical case is given and, moreover, an analogy with behaviour of solutions of...

Blow-up results for some reaction-diffusion equations with time delay

Hongliang Wang, Yujuan Chen, Haihua Lu (2012)

Annales Polonici Mathematici

We discuss the effect of time delay on blow-up of solutions to initial-boundary value problems for nonlinear reaction-diffusion equations. Firstly, two examples are given, which indicate that the delay can both induce and prevent the blow-up of solutions. Then we show that adding a new term with delay may not change the blow-up character of solutions.

Currently displaying 241 – 260 of 1830