LINEAR DIFFERENTIAL EQUATIONS WITH UNBOUNDED DELAYS AND A FORCING TERM

The paper discusses the asymptotic behaviour of all solutions of the differential equation ẏ(t)=−a(t)y(t) +∑ni=1 bi(t)y(τi(t)) + f (t), t ∈ I = [t0,∞), with a positive continuous function a, continuous functions bi, f , and n continuously differentiable unbounded lags. We establish conditions under which any solution y of this equation can be estimated by means of a solution of an auxiliary functional equation with one unbounded lag. Moreover, some related questions concerning functional equations are discussed as well.


Introduction
In this paper, we study the problem of the asymptotic bounds of all solutions for the delay differential equation ẏ(t) = −a(t)y(t) + n i=1 b i (t)y τ i (t) + f (t), t ∈ I = t 0 ,∞ , ( where a is a positive continuous function on I; b i , f are continuous functions on I, τ i are continuously differentiable functions on I fulfilling τ i (t) < t, 0 < τi (t) ≤ λ i < 1 for all t ∈ I and τ i (t) → ∞ as t → ∞, i = 1,...,n.
The prototype of such equations may serve the equation with proportional delays ẏ(t) = −ay(t) where a > 0, b i = 0, 0 < λ i < 1, i = 1,...,n, are real scalars.There are numerous interesting applications for (1.2) and its modifications, such as collection of current by the pantograph head of an electric locomotive, probability theory on algebraic structures or partition problems in number theory.Various special cases of (1.2) have been studied because of these applications, as well as for theoretical reasons (see, e.g., Bereketoglu and Pituk [1], Lim [11], Liu [12], or Ockendon and Taylor [15]).
The study of these differential equations with proportional delays turned out to be the useful paradigm for the investigation of qualitative properties of differential equations with general unbounded lags.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].
In this paper, we combine standard methods from the theory of functional differential equations and some results of the theory of functional equations and difference equations to analyze the asymptotic properties of all solutions of (1.1).The main results are formulated in Sections 3 and 4. In Section 3, we derive the asymptotic estimate of all solutions of (1.1).Section 4 discusses some particular cases of (1.1) and improves the above derived estimate for these special cases.Both sections also present the illustrating examples involving, among others, (1.2).
In the sequel, we introduce the notion of embeddability of given functions into an iteration group.This property will be imposed on the set of delays {τ 1 ,...,τ n } throughout next sections.Definition 2.1.Let ψ ∈ C 1 (I −1 ), ψ > 0 on I −1 .Say that {τ 1 ,...,τ n } can be embedded into an iteration group [ψ] if for any τ i there exists a constant d i such that (2.1) Remark 2.2.The problem of embeddability of given functions {τ 1 ,...,τ n } into an iteration group [ψ] is closely related to the existence of a common solution ψ to the system of the simultaneous Abel equations The complete solution of these problems have been described by Neuman [14] and Zdun [16].These papers contain conditions under which (2.1) holds for any τ i , i = 1,...,n (see also [10,Theorem 9.4.1]).We only note that the most important necessary condition is commutativity of any pair τ i , τ j , i, j = 1,...,n.Notice also, that if τ i are delays, then d i must be positive.

The asymptotic bound of all solutions of (1.1)
The aim of this section is to formulate and prove the asymptotic estimate of all solutions of (1.1).We assume that all the assumptions imposed on a, b i , τ i , and f in Section 1 are valid.
Proof.The substitution where " " stands for d/ds, h(s) = ψ −1 (s), and ..,n.This form can be rewritten as where (3.7) we can rewrite (3.7) as (3.9) From here, we get a(u)du where K 2 = K 1 /K.Using the assumptions imposed on a and τi , we can estimate the integral s0 a(u)du}ds as (for a similar situation see also [4]).Hence, a(u)du where M * k := max(M k ,K 2 ), κ := min(ω,γ − α − β) > 0, and N > 0 is a constant large enough.Since s * ∈ J k+1 was arbitrary, Now, the boundedness of (M * k ) as k → ∞ implies via substitution (3.2) the asymptotic estimate (3.1).Remark 3.2.This remark concerns the possible extension of our results to differential equations with delays intersecting the identity at the initial point t 0 .These equations form a wide and natural class of delay differential equations (see the following examples) and have many applications (some of them have been mentioned in Section 1).Since we are interested in the behaviour at infinity, it is obvious that the main notions and results of this paper can be easily reformulated to this case.

Some particular cases of (1.1)
In this section, we first consider (1.1) in the homogeneous form Using a simple modification of the proof of Theorem 3.1, we improve the conclusion of this theorem for the case of (4.1).We assume that all the assumptions of Theorem 3.1 are valid (the assumptions on f are missing, of course).Using the same notation as in Theorem 3.1, we have the following theorem.
Theorem 4.1.Let y be a solution of (4.1).Then and this implies the validity of (4.2).Now, we consider (1.1) in another special form ẏ(t) = −ay(t) + by τ(t) + f (t), t ∈ I, ( where Under these assumptions on τ, there exists a function ψ ∈ C 1 (I), ψ > 0 on I such that (for this and related results concerning (4.6) see, e.g., [10]).Then applying Theorem 3.1 to (4.5), we can easily deduce that the property holds for any solution y of (4.5).The asymptotic behaviour of (4.5) has been studied in [3] (4.9) It is easy to see that relations (4.9) yield sharper estimates of solutions than (4.7).On the other hand, we emphasize that the proof technique used in [3] is effective just for (4.5) and cannot be applied to more general equation (1.1).In the final part of this paper, we propose a simple way on how to extend the conclusions of Theorem 4.2 to some equation (4.5) with nonconstant coefficients.To explain the main idea, we consider (4.5),where the delayed argument is a power function.
where a > 0, b = 0, 0 < λ < 1 are constants and f ∈ C 1 ([1,∞)) fulfils the properties The corresponding Abel equation (4.6) has the form and admits the function ψ(t) = loglogt as a solution with the required properties.Substituting this ψ into assumptions and conclusions of Theorem 4.2, we obtain the following result, where σ is given by (4.8), if y is a solution of (4.10), then (s) exp − γd i h (s)exp γs + )du f h(s) exp{−γs}h (s)ds.

Example 4 . 3 .
We consider the delay equation ẏ functions {λ 1 t,...,λ n t} can be embedded into an iteration group [ψ].Indeed, ≡ 0 on I.Moreover, the inequality (3.8) can be replaced by Now if the forcing term in(4.16)fulfils the required asymptotic properties, then applying Theorem 4.2 to (4.16) and substituting this back into (4.15),we get that relations (4.13) are valid for any solution y of (4.14).Remark 4.4.Following Example 4.3, we can extend asymptotic estimates (4.13) also to some other equations of the form ẏ(t) = − φ(t) ay(t) − by t λ + f (t), t ≥ 1, (4.17) where ϕ ∈ C 1 ([1,∞)) and φ > 0 on [1,∞).If we introduce the change of variables Now if the delayed argument and the forcing term in (4.19) fulfil the assumptions of Theorem 4.2, then we can apply this theorem to (4.19) and via substituting (4.18) obtain the validity of (4.13) for any solution y of (4.17).