Asymptotic behaviour of solutions of some differential equations with an unbounded delay

J. Cermák, Technical University of Brno, Brno, Czech Republic

E. J. Qualitative Theory of Diff. Equ., Proc. 6'th Coll. Qualitative Theory of Diff. Equ., No. 2. (2000), pp. 1-8.

Abstract: We investigate the asymptotic properties of all solutions of the functional differential equation
$$\dot{x}(t)=p(t)[x(t)-kx(t-\tau (t))]+q(t),\qquad t\in I=[t_0,\infty),$$
where $k\ne 0$ is a scalar and $\tau (t)$ is an unbounded delay. Under certain restrictions we relate asymptotic behaviour of solutions $x(t)$ of this equation to the behaviour of a solution $\varphi (t)$ of the auxiliary functional nondifferential equation
$$\varphi (t)=|k|\,\varphi (t-\tau (t)),\qquad t\in I.$$

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