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

Abstract

We investigate the asymptotic properties of all solutions of the functional differential equatioṅ x(t) = p(t)[x(t) − kx(t − τ (t))] + q(t), t ∈ I = [t 0 , ∞), where k = 0 is a scalar and τ (t) is an unbounded delay. Under certain restrictions we relate the asymptotic behaviour of the solutions x(t) of this equation to the behaviour of a solution ϕ(t) of the auxiliary functional nondifferential equation ϕ(t) = |k| ϕ(t − τ (t)), t ∈ I.

The Schröder equation and asymptotic properties of linear delay differential equations

“…Remark. As we have already mentioned in Section 1, equation (1.1) with a positive coefficient at x(t) was the subject of asymptotic investigations in [3]. It is natural to expect that the behaviour of the solutions at infinity described in [3] is quite different from that described in the previous Theorem.…”

“…As we have already mentioned in Section 1, equation (1.1) with a positive coefficient at x(t) was the subject of asymptotic investigations in [3]. It is natural to expect that the behaviour of the solutions at infinity described in [3] is quite different from that described in the previous Theorem. Indeed, if p is nondecreasing and q fulfils some asymptotic bounds, then for any solution x of (1.1) (with the positive sign at x(t)) there exists a (possibly zero) constant L ∈ R such that…”

“…(see [3,Lemma 1]). Of course, the exponential behaviour (if L = 0) of solutions cannot be estimated via terms occuring in the previous Theorem.…”

“…Our aim is to establish asymptotic conditions on the forcing term q under which results of this type remain valid also for equation (1.1). Note that equation (1.1) with a positive coefficient at x(t) has been studied in [3], where the asymptotics of solutions have been determined by means of the behaviour of a solution of the auxiliary differential equatioṅ…”

Linear differential equations with unbounded delays and a forcingterm

“…Some results of the above-cited papers have been generalized in this direction by Heard [7], Makay and Terjéki [13], and in [2,3,4]. For further related results on the asymptotic behaviour of solutions, see, for example, Diblík [5,6], Iserles [8], or Krisztin [9].…”