Asymptotic formulas for a scalar linear delay differential equation

The linear delay differential equation x′(t) = p(t)x(t− r) is considered, where r > 0 and the coefficient p : [t0, ∞) → R is a continuous function such that p(t) → 0 as t → ∞. In a recent paper [M. Pituk, G. Röst, Bound. Value Probl. 2014:114] an asymptotic description of the solutions has been given in terms of a special solution of the associated formal adjoint equation and the initial data. In this paper, we give a representation of the special solution of the formal adjoint equation. Under some additional conditions, the representation theorem yields explicit asymptotic formulas for the solutions as t→ ∞.


Introduction
Consider the delay differential equation where r > 0 and p : [t 0 , ∞) → R is a continuous function.The initial value problem associated with (1.1) has the form x(t) = φ(t), where t 1 ≥ t 0 and φ : [t 1 − r, t 1 ] → R is a continuous function.Recently, under the smallness condion Corresponding author.Email: gyori@almos.uni-pannon.huwe have given an asymptotic description of the solution of the initial value problem (1.1) and (1.2) in terms of a special solution of the formal adjoint equation y (t) = −p(t + r)y(t + r). (1.4) We have shown the following theorem (see Theorems 3.1-3.3 in [10]).
Theorem 1.1.Suppose (1.3) holds.Then up to a constant multiple the adjoint equation (1.4) has a unique solution y on [t 0 , ∞) which is positive for all large t and satisfies lim sup t→∞ y(t + r) y(t) < ∞. (1.5) Furthermore, if x is the solution of the initial value problem (1.1) and (1.2), then where c is a constant given by c = φ(t 1 )y(t 1 ) + t 1 t 1 −r p(s + r)φ(s)y(s + r) ds. (1.7) In the sequel, the solution y of the adjoint equation described in Theorem 1.1 will be called a special solution of Eq. (1.4).
A close look at the proof of Theorem 3.1 in [10] shows that the special solution of the adjoint equation y has the following additional properties: if t 1 ≥ t 0 is chosen such that We emphasize that (1.6) gives a genuine asymptotic representation of the solutions of Eq. (1.1) in the sense that there exists a solution x of (1.1) for which the constant c in (1.6) is nonzero.Indeed, if t 1 is chosen such that (1.8) is satisfied, then for the solution x of (1.1) with initial data (1.2) defined by we have (by (1.7)), the second and the last inequality being a consequence of (1.10) and (1.8), respectively.Our previous study [10] was motivated by the Dickman-de Bruijn equation (see [1,2,5]) for which the special solution of the associated adjoint equation can be given explicitly by y(t) = t for t ≥ 1.Thus, in this case (1.6) leads to the explicit asymptotic representation For similar qualitative results, see [3,4,[6][7][8] and the references therein.
In contrast with the Dickman-de Bruijn equation (1.11), in most cases we do not know an explicit formula for the special solution of the adjoint equation (1.4).Therefore the purpose of the present paper is to describe the special solution of the adjoint equation (1.4) in terms of the coefficient p and the delay r.In Section 2, we prove a new representation theorem for the special solution of the adjoint equation (1.4) (see Theorem 2.1 below).In Section 3, in Theorem 3.1, we show that under some additional conditions the representation theorem yields explicit asymptotic formulas for the solutions of the linear delay differential equation (1.1).

Representation of the special solution of the adjoint equation
To simplify the calculations instead of (1.3) we will assume the slightly stronger condition This implies that if t 1 ≥ t 0 is sufficiently large, then Clearly, condition (2.2) implies (1.8).Therefore, under condition (2.2), the special solution y of the adjoint equation has properties (1.9) and (1.10).
In order to formulate our main representation theorem, we need to introduce some auxiliary functions.Define and 2) is satisfied, then the unique special solution y of the adjoint equation (1.4) with property y(t 1 ) = 1 is given by ) the function series on the righ-hand side being uniformly convergent on [t 1 , ∞).
the last equality being a consequence of conclusion (2.9) of Lemma 2.2.This proves that (2.10) holds for all n.
Now we are in a position to give a proof of Theorem 2.1.
Proof.We will show that the series (2.6) converges uniformly on [t 1 , ∞).First we prove by induction that for every positive integer k, where q is defined by (2.2).By virtue of (2.2) and (2.3), we have Thus, (2.11) holds for k = 1.Now suppose that (2.11) holds for some positive integer k.Then for s ≥ t ≥ t 1 , This proves that (2.11) holds for all k.From (2.11), we find that for every positive integer k, From this and the inequality we obtain for every positive integer k, Since qre < This proves that (2.15) holds for all k.From (2.15), we obtain for every positive integer n, In view of (2.13), the last inequality implies that β n (t, t) → 0 uniformly on [t 1 , ∞) as n → ∞.
Finally letting n → ∞ in conclusion (2.10) of Lemma 2.3, we obtain that the special solution y of Eq. (1.4) satisfies the ordinary differential equation where σ is defined by (2.6).Since y(t 1 ) = 1, this implies that y has the form (2.5) and the proof of Theorem 2.1 is complete.

Explicit asymptotic formulas
From Theorems 1.1 and 2.1, we can deduce explicit asymptotic formulas for the solutions of Eq. (1.1).
Theorem 3.1.Suppose that there exists a positive monotone decreasing function a and Then for every solution x of Eq. (1.1) there exists a constant γ such that where σ n is the n th partial sum of the function series (2.6), Moreover, the asymptotic formula (3.3) is genuine in the sense that there exists a solution x of (1.1) for which the constant γ in (3.3) is nonzero.
Proof.First we prove by induction that under the hypotheses of the theorem for all positive integer k, By virtue of (2.3) and (3.1), we have for s ≥ t ≥ t 0 , where the last inequality is a consequence of the monotonicity of a. Thus, (3.5) holds for k = 1.Now suppose that (3.5) holds for some positive integer k.Then we have for s ≥ t ≥ t 1 , the last inequality being a consequence of the monotonicity of a.This proves that (3.5) holds for all k.From (3.5), we find that for all positive k, From this, using inequality (2.12) and taking into account that a is monotone decreasing, we obtain for all k, Choose q > 0 such that qre < 1.Since a is monotone decreasing and (3.2) holds, it follows that a(t) → 0 as t → ∞.Therefore there exists t 1 ≥ t 0 such that sup By the application of Theorem 2.1, we conclude that the special solution y of the adjoint equation (1.4) with property y(t 1 ) = 1 has the form (2.5).This, combined with Theorem 1.1, implies that every solution x of Eq. (1.1) satisfies the asymptotic relation where c is a constant depending on x.Moreover, as shown in Section 1, there exists a solution x of Eq. (1.1) for which c > 0. For t ≥ t 1 , define We will show that  where η is a constant depending on x.Moreover, there exists a solution x of Eq. (3.22) for which η = 0.
Note that the asymptotic representation (3.27) implies the following interesting stability criteria for Eq.(3.22) (see Corollary 3.2 in [8]).(iii) The zero solution of Eq. (3.22) is stable, but it is not asymptotically stable if and only if r = kπ for some k ∈ Z + .
.10)From (3.6) and (3.7), we obtain for t ≥ t 1 , (t) dt converges.Since σ n = σ − ρ n , we have for t ≥ t 1 , Thus, in this case the constant γ in the asymptotic relation(3.3) is always zero.Therefore if hypothesis (3.2) is not satisfied, then (3.3) in general does not give a genuine asymptotic description of the solutions as t → ∞.Now we prove (3.13).Using the facts that p is negative and |p| is monotone decreasing, it follows by easy induction that for all positive k,α k (t, s) ≥ r k−1 |p(s + kr)| k > 0 whenever s ≥ t ≥ t 0 .(3.14)As noted in the proof of Theorem 3.1, if t 1 ≥ t 0 is chosen such that (3.7) is satisfied, then for every solution x of (1.1) the asymptotic formula (3.8) holds.If ρ n is defined by (3.9), then by virtue of (3.14), we have for t ≥ t 1 , ρ n (t) ≥ α n+1 (t, t) ≥ r n |p(t + (n + 1)r)| n+1 .(s, s) ds −→ κ, t → ∞, where κ = κ 2 κ 3 > 0. From this and (3.23), by the application of Theorem 3.1, we conclude that every solution x of Eq. (3.22) has the form x(t) = t − sin r 2 (η + o(1) as t → ∞, (3.27) n (s) ds −→ d = c exp − n (s) ds .Clearly, if c is nonzero, then so is d and hence γ.This completes the proof of the theorem.Remark 3.2.To illustrate the importance of hypothesis (3.2) in Theorem 3.1 condsider Eq. (1.1), where p : [t 0 , ∞) → (−∞, 0) is a negative monotone increasing function which tends to zero as t → ∞.Clearly, in this case condition (3.1) holds with a = |p|.(Anexample of such a p is the function p(t) = − ln −1 t defined for t ≥ 2.) We will show that if σ n has the meaning from Theorem 3.1, then for every positive integer n, x(t) exp