Explicit Integral Criteria for the Existence of Positive Solutions of First Order Linear Delay Equations

It is well known that the linear differential equation ˙ x(t) + p(t)x(t − τ(t)) = 0 a positive solution on [t 0 , ∞) if an explicit criterion of the integral type t t−τ(t) p(s)ds ≤ 1 e holds for all t ∈ [t 0 , ∞). In this paper new integral explicit criteria, which essentially supplement related results in the literature are established. For example, if, for t ∈ [t 0 , ∞) and a fixed µ ∈ (0, 1), the integral inequality t t−τ/2 p(s) ds ≤ 1 2e + µτ 3 96t 3 e holds, then there exists a t * 0 ≥ t 0 and a positive solution x = x(t) on [t * 0 , ∞). Examples illustrating the effectiveness of the results are given.


p(s)ds ≤ 1 e
holds for all t ∈ [t 0 , ∞).In this paper new integral explicit criteria, which essentially supplement related results in the literature are established.For example, if, for t ∈ [t 0 , ∞) and a fixed µ ∈ (0, 1), the integral inequality

Introduction
The purpose of the paper is to derive explicit integral criteria for the existence of eventually positive solutions to the equation ẋ(t) + p(t)x(t − τ(t)) = 0, ( with t ≥ t 0 ∈ R, in terms of inequalities of the integral of the coefficient p (without loss of generality, we will assume t 0 sufficiently large throughout the paper to ensure that the performed computations are well-defined) where p : [t 0 , ∞) → (0, ∞) is a continuous function.
A solution to (1.1) is defined as follows: a continuous function x : [t * − r, ∞) → R is called a solution of (1.1) corresponding to t * ∈ [t 0 , ∞) if x is differentiable on [t * , ∞) (the derivative at t * is regarded as the right-hand derivative) and satisfies (1.1) for all t ≥ t * .A solution of (1.1) corresponding to t * is called oscillatory if it has arbitrarily large zeros.Otherwise, it is called non-oscillatory.A non-oscillatory solution x of (1.1) corresponding to t * is called positive (negative) if x(t) > 0 (x(t) < 0) on [t * − r, ∞).A solution x of (1.1) corresponding to t * is called eventually positive (eventually negative) if there exists t * * > t * such that x(t) > 0 (x(t) < 0) on [t * * , ∞).
Equation (1.1) often serves, due to its simple form, as an equation prototype for testing and comparing new results.But equation (1.1) itself has interesting applications as well.It is, for example, well-known in number theory that what is called the Dickman-de Bruijn function (we refer to [36,38] and to the references therein) is a positive solution of the initial problem In [22, p. 226] equation (1.1), in the case of the coefficient and the delay in (1.1) being constant, p(t) = p > 0 and τ(t) = τ > 0, i.e. ẋ(t) + px(t − τ) = 0, t ≥ t 0 (1.2) models the amount of salt (expressed by a positive solution) in the brine in a tank diluted by fresh water.The same equation is used in an example of water temperature regulation by a showering person in [32, p. 74].A well-known example of type (1.1) equation ẋ(t) + 2te 1−2t x(t − 1) = 0 with a solution x(t) = e −t 2 illustrates the fact that linear equations with delay can have positive solutions decreasing for t → ∞ to zero faster than an arbitrary exponential function e −αt where α > 0 (see, e.g.[31, p. 97]).
It is well-known that either there exists an eventually positive solution of (1.1) or every solution of (1.1) is oscillatory.In the literature, by a critical case of the coefficient p in (1.1) is usually understood a boundary for p separating, in a sense, both the above mentioned asymptotically different qualitative cases of behavior of solutions to (1.1).We can give an explanation in the case of equation (1.2).In such a case, it is easy to show that there exists a positive solution if pτ ≤ 1/e and that all solutions oscillate if pτ > 1/e, the value 1/e is called the critical value.
In the paper, we develop some new explicit integral criteria related to the well-known classical sharp integral criterion ).Now we give a short overview of known results stating the existence of a positive solution to (1.1).

Implicit criterion
The following well-known implicit criterion (with conditions adapted for (1.1)) on the existence of positive solutions is often cited in the literature.This criterion can be found, e.g., in [27, Theorem 1, Assertion 7 and Corollary 2.1] and also in [1,2] and [24,Theorem 2.1.4.].Inequality (1.4) is of considerable importance since it often plays a crucial role in the process of deriving explicit criteria of positivity.

Explicit criteria
Some results cited below are formulated explicitly in terms of inequalities for the coefficient p or in terms of integrals containing p.These results deal with the critical case and are sharp (non-improvable) in various senses (often explained in the original papers).E.g., positive solutions might not exist if the cited inequalities are subject to certain small perturbations.
In some of the inequalities below appears what is called the iterated logarithm.We define iterated logarithms of k-th order as ln k t := ln ln . . .ln k t, k ≥ 1, t > exp k−2 1 , ln 0 t := t and the iterated exponential exp k t := ( exp(exp(. . .exp k t ))), exp 0 t := t, exp −1 t := 0 is used to determine the domain of the iterated logarithm.

Point-wise criteria
In [23] it is assumed that p(t) = 1/e + a(t), τ(t) = 1 and t 0 = 1.Then, the equation for all sufficiently large t [23, Theorem 3].This result is improved in [20] as follows.If for all sufficiently large t, then (1.5) has a positive solution.A further generalization is given in [10], where it is proved that, for the existence of a positive solution to (1. in [17], it is proved that, for the existence of an eventually positive solution of (1.1), it is sufficient if an integer k ≥ 0 exists such that Moreover, in [3], it is showed that, if (1.8) holds and 0 ≤ τ(t) ≤ r for t → ∞, then (1.1) has an eventually positive solution.We finish this short overview by including a general result published in [3].Let 1/τ(t) be a locally integrable function and If there exists a δ ∈ (0, ∞) such that and, for a fixed integer k ≥ 0, where then there exists an eventually positive solution of (1.1).

New explicit integral criteria
Substituting (1.9) into (1.4),where ω : Then there exists a positive solution x = x(t) of (1.1) on [t 0 , ∞) satisfying the inequality Theorem 2.1 is used in the proof of the following theorem.
Proof.For the left-hand side L of inequality (2.1), we get p(s) ds.Now, obviously, an estimate of the right-hand side R of inequality (2.1), utilizing (2.3), is Inequality (2.1) holds and from Theorem 2.1 the proof of Theorem 2.2 is complete.Now we use Theorem 2.2 to get an easily verifiable explicit criterion.

Further positivity criteria
In this part, some further positivity criteria are derived.First we generalize Theorem 2.2 in the case that the interval [t − τ(t), t] is divided by several points.
Proof.In the proof, we apply Theorem 2.1 again.For the left-hand side L of inequality (2.1) we get, using (3.1), p(s)e −ω(s) ds ≤ e −ω(t−θ 1 (t)τ(t)) p(s)e −ω(s) ds = L, inequality (2.1) holds, and from Theorem 2.1 the proof of Theorem 3.1 is complete.Now we use Theorem 3.1 to get an easily verifiable explicit criterion when the interval [t − τ, t], t ≥ t 0 is divided into n subintervals.It is necessary to underline that it is assumed that n > 2, i.e.Theorem 3.2 below cannot be reduced to Theorem 2.3 and both theorems are independent.It is a surprising fact that the proof of Theorem 3.2 is even simpler than that of proof of Theorem 2.3 (because the terms of the third order of accuracy in the asymptotic decomposition are not necessary) and, simultaneously, the function α satisfies an estimation (3.3) below which is weaker than (2.6) in Theorem 2.3.Theorem 3.2.Let n > 2 be an integer and for all t ≥ t 0 > 0 and i = 1, . . ., n, and a constant M.
Proof.Put in (3.1) ω(t) = τ(t)/(2t) and θ i (t) = (n − i)/n, i = 1, . . ., n.Then, (3.1) equals (below by L * 1 and R * 1 the left-hand and the right-hand sides of (3.1) are denoted) p(s) ds Obviously, due to the boundedness of τ(t), due to (3.2) and, combining both properties (3.5), (3.6), we have for positive integers m and s and i = 1, . . ., n − 1.By (3.5)-(3.7), it is possible to asymptotically decompose exponential functions in (3.4).This is the next step.For i = 1, . . ., n, we get exp −τ(t − ((n Then, utilizing (3.3) and (3.8) we can estimate the left-hand side of (3.4), is sufficient.Let us transform the second sum in (3.9).We get where • is the floor function.Since, by (3.2), and (3.9) will hold if Now we apply inequality (3.3).Then, inequality (3.10) will hold if Since, by (3.2), We have n > 2 and µ ∈ (0, 1) so that the last inequality is obvious and If n = 2 (which is not allowed in Theorem 3.2), this term disappears and the sign will be determined by expressions of order higher than τ 2 (t)/t 2 .Such an approach and detailed analysis is carried out in the proof of Theorem 2.3.
A minor modification in the proof of Theorem 3.2 results in the following statement.
Asymptotic analysis of all the terms on the left-hand side of (3.16) leads to a conclusion that its sign is determined by the sum of two negative terms (recall that 1 − µ > 0) because all the remaining terms are of an asymptotically higher order than at least one these two negative terms.Further, we can proceed as in the proof of Theorem 3.
Proof.From (4.3), we get With this modification against the original proof of Theorem 3. Let us formulate some open problems for future research.Although, in the paper, we provided several new explicit integral criteria for the existence of a positive solution x = x(t) of (1.1) and we demonstrated that our criteria are independent of the previously known results, unfortunately, our approach could not, in its present form, improve the classical criterion (1.3)Similarly, tracing carefully the present results, open problems connected with Theorems 2.3, 2.4, 3.2 and 3.4 can be formulated in the case of a variable delay.
In the paper we also pointed out the difference between Theorem 2.3 and Theorem 3.2.A question arises, how the method used can be improved to get a better estimate of the function α in Theorem 2.3.Therefore, the following problem described below is another challenge for future investigation.then there exists a t * 0 ∈ [t 0 , ∞) and a positive solution x = x(t) of (1.1) on [t * 0 , ∞).In Theorems 2.3, 2.4, 3.2, and 3.4, inequalities (2.4), (2.16), (3.2), and (3.13) were used.These inequalities are valid if delay is nonincreasing (the case of inequalities (2.4), (3.2)) or decreasing (such possibility is admitted in all four inequalities).The last but not least task is whether similar results on the existence of positive solutions can be derived if the delay is nondecreasing or increasing.Finally, we refer to papers [5,6,8,39] where similar problems of the behavior of solutions of delayed equations are treated.
Comparing inequality(2.6)inTheorem 2.3 with inequality (2.17) in Theorem 2.4, we conclude that these theorems are independent.Let us illustrate this remark by two examples with different delays (and note that neither point-wise criteria mentioned in 1.2.1 nor integral criterion (1.3) are applicable).