Convergence in Asymptotically Autonomous Functional Differential Equations

Abstract

In this paper, we consider linear and nonlinear perturbations of a linear autonomous functional differential equation which has infinitely many equilibria. We give sufficient conditions under which the solutions of the perturbed equation tend to the equilibria of the unperturbed equation at infinity. As a consequence, we obtain sufficient conditions for systems of delay differential equations to have asymptotic equilibrium.

Convergence of the solutions of the equation y˙(t)=β(t)[y(<

“…But recently, the problem of asymptotic convergence or divergence of solutions of linear delayed equations has received much attention. Let us mention at least investigations [1][2][3][4][5][6][7][8][9][10][11]. Comparing the known investigations with the results presented we can see that our results can be applied when we deal with the so called critical case and give more exact sufficient conditions for this case.…”

“…(1) with solutions of an auxiliary inequality which formally copies Eq. (1). At first, we prove that, under certain conditions, Eq.…”

“…Then we extend this statement to all the solutions of Eq. (1). Moreover, in the general case the convergence of all solutions is characterized by the existence of a strictly increasing bounded solution.…”

Exponential solutions of equation ẏ(t)=β(t)[y(t−δ)−y(t−τ)]

“…In the paper presented we are trying to represent solutions of Eq. (1) by means of exponential-like functions exp t t 0 −τε (s)β(s) ds (2) with a continuous functionε : I −1 \ {t 0 } → (0, 1). We call representation (2) exponential representation (being aware of that, e.g., for functionsε close to 0, this form can give a different kind of a function than just exponential ones).…”

On existence of solutions of a neutral differential equation with deviating argument

“…The problem (1)- (2) was considered in [4,11] (see also [1,2,3,6,7,8,9,12,13] under strong assumptions. Authors obtained their results by using two-component measure of noncompactness in the Banach space C p (R + ) consisting of all continuous functions on R + and tempered by p.…”