1. Introduction and Preliminary Results
Consider the following linear differential equation with variable delay:
where
and
are continuous functions, such that
(as
). Note that
is not assumed to be nondecreasing. Let
and note that
holds. Then, a continuous function
is called a
solution of Equation (
1), if it is continuously differentiable on
and satisfies Equation (
1) there.
Such equations, and, in general, delay differential equations with either constant or variable delay arise naturally in a multitude of models from biology, physics, engineering, chemistry and economy. For an extensive introduction to the theory of delay differential equations, we refer to the books [
1,
2], whereas for more on their applications we recommend the reader to study [
3,
4].
This paper is concerned with the oscillatory behaviour of Equation (
1). By convention, a solution is called
oscillatory if it has arbitrary large zeros and is
nonoscillatory otherwise. Results on oscillation of retarded first order equations already appeared in the works of Johann Bernoulli [
5]. The first systematic study of oscillatory and nonoscillatory behaviour of Equation (
1) goes back to Myshkis [
6]. He showed that, in case the functions
and
p are bounded, then
implies that all solutions of Equation (
1) are oscillatory, whereas condition
guarantees the existence of a nonoscillatory solution.
Since then, the question of oscillation has received much attention and many results have been published providing sufficient conditions guaranteeing that all solutions are oscillatory and others that establish the existence of a nonoscillatory solution. For more details, we refer the interested reader to monographs [
7,
8,
9] and to the survey papers [
10,
11]. Here, we only point out some results that are most relevant from our perspective.
Ladas, Lakshmikantham and Papadakis [
12] proved that all solutions of Equation (
1) are oscillatory, provided
The following important contribution is due to Koplatadze and Chanturija [
13]. For the proof, see also e.g., Theorem 2.1.1 of [
9].
Theorem 1 - (i)
Ifthen all solutions of Equation (
1)
are oscillatory. - (ii)
Ifor, more generally, ifthen Equation (
1)
has a nonoscillatory solution.
After these central results, many works have focused on filling the gap between Conditions (
2) and (
3), as well as between the necessary and the sufficient conditions given by Theorem 1 and Condition (
4). For more on such results, see, e.g., the recent survey by Moremedi and Stavroulakis [
10].
It is worth mentioning that, in case the functions
and
p are constant, then both Conditions (
5) and (
2) reduce to condition
, which is in this case not only sufficient, but—in view of Inequality (
3)—also necessary for the oscillation of all solutions. Another immediate corollary of Theorem 1 is that, if
is constant
, and
p is
-periodic, then
is constant and Condition (
7) is sharp.
Motivated by these facts, Pituk [
14] recently proved that, for constant delay
, there is a class of functions
p, for which the ‘almost necessary’ condition
is sufficient for the oscillation of all solutions of Equation (
1). More precisely, he showed in Theorem 1 of [
14] that, if
p is slowly varying at infinity with
, then
implies that all solutions of Equation (
1) are oscillatory, where a function
is called
slowly varying at infinity if, for every
,
In a subsequent paper, Pituk, Stavroulakis, and the present author [
15] generalized the above result and gave a class of functions
p—broader than
-periodic—for which Condition (
6) is ‘almost sharp’. More precisely, the following theorem was proved.
Theorem 2 ([
15])
. Let the function τ in Equation (1) be constant, and function p be nonnegative, bounded and uniformly continuous. Assume further that the function is slowly varying at infinity. Then,imply that all solutions of Equation (1) are oscillatory. The purpose of this paper is to show that Theorem 2 remains valid in case of variable delay, provided is uniformly continuous and bounded. The proof is similar to that of Theorem 2; nevertheless, some technical difficulties also arise due to the variable delay.
In the next section, we present our main theorems and give some hints to support applicability of the results. Then, in
Section 3, we provide an illustrative example.
Section 4 is devoted to conclusions.
2. Results
The following theorem is our main result.
Theorem 3. For some positive numbers M and κ, let and be uniformly continuous functions, and suppose that the functionis slowly varying at infinity. Then,imply that all solutions of Equation (1) are oscillatory. Before we prove the theorem, we make some comments, mainly to support applicability of the result.
From Theorem 1, it is apparent that condition
is necessary for the oscillation of all solutions, so Theorem 3 is sharp in this sense. Example 9 of [
15] showed that the slowly varying assumption is important: even in the constant delay case, the theorem does not hold if we omit that assumption.
We remark that uniform continuity of p and are guaranteed, if they are globally Lipschitz continuous, which is the case if they are differentiable with their derivatives bounded on .
Let us also devote some comments to functions that are slowly varying at infinity—we shall call them slowly varying for brevity.
The class of slowly varying functions was studied already by Karamata [
16] in a multiplicative form. For more information about slowly varying functions and their characterization, we refer the reader to the monograph by Seneta [
17]. In particular, for the relation between the two terminologies, see the remark below Theorem 1.2 in Chapter 1 of [
17].
Here, let us mention only one characterization of slowly varying functions given by Pituk [
14] (in the additive form, see Formula (
9)): a continuous function
is slowly varying if and only if there exists
, such that
f can be written in the form
where
is a continuous function which tends to some finite limit as
, and
is a continuously differentiable function for which
holds.
The next lemma will be essential in our proof.
Lemma 1 ([
13]).
Suppose that is a continuous function satisfyingIf x is an eventually positive solution of Equation (1), then, for all sufficiently large T, Proof of Theorem 3. Assume to the contrary that x is an eventually positive solution and all assumptions of the theorem hold (if the solution x is eventually negative, then take the solution ).
By virtue of Lemma 1, there exists
such that
holds for all
and
Then, there exists a sequence
, such that
and
Let us introduce the following sequence of functions:
Then, applying (
1) leads to the equation
Now, we would like to pass to the limit by applying the Arzelà–Ascoli theorem for the above sequences of functions , and , hence we need to establish their uniform boundedness and equicontinuity. Uniform boundedness, respectively equicontinuity of and follow from the boundedness, respectively uniform-continuity of functions p and .
It remains to check these properties for
. For this, note that by virtue of Equation (
1) and Equation (
14) we obtain that the inequality
holds for all
and
. This immediately implies
As
is positive on
, we obtain inequalities
Taking into account that
for all
, we obtain that
holds for all
and
. Now, Inequalities (
20) and (
18) imply that
and
are uniformly bounded on
. Furthermore, the uniform boundedness of
yields that functions
are globally Lipschitz continuous with a common Lipschitz constant, and consequently
is uniformly equicontinuous.
In view of the above, by the Arzelà–Ascoli theorem, we may assume (by passing to a subsequence without changing notation) that the limits
exist and are continuous on
, and the convergence is uniform on every bounded subinterval of
. Note that
also holds for all
and
.
Furthermore, from Equation (
16), together with the uniform continuity of functions
p and
and the uniform equicontinuity of
, we obtain that
is also equicontinuous on
. Recall that the sequence
is uniformly bounded on
. Hence, according to the Arzelà–Ascoli theorem, we may assume (after passing to a subsequence if necessary) that the limit
exists for all
, and the convergence is uniform on all bounded subintervals of
. This combined with the fact that
yields (see, e.g., Theorem 7.17 of [
18]) that
holds for all
. By virtue of Equation (17),
is satisfied for all
. From Equation (
21) and the (uniform) equicontinuity of
, one can easily derive that
holds for all
. Thus, Inequality (
22) impies that
y is a positive solution of equation
As a final step, we will apply Theorem 1 (i) to show that every solution of Equation (
24) is oscillatory, which is a contradiction. Thus, we need to verify that Equation (
24) fulfils the hypotheses imposed on Equation (
1) and that Inequality (
5) holds.
First, observe that and for all follow immediately from their definitions and from the assumptions on p and , respectively. Note that we have not yet shown that is positive for all t.
Next, we prove that Inequality (
5) is satisfied. For this, let us fix
and note that, since
converges uniformly to
q on the interval
, we obtain
The functions
are uniformly bounded, and
, as
, so the limit of the last integral vanishes. This in turn leads to
Here, the last inequality and the last equality hold by assumption, whereas the last but one equality follows from the slowly varying property of
A. Hence,
is constant
B, and thus Inequality (
5) holds.
The only condition that still needs to be verified is that
is positive for all
. Notice that this follows immediately from the above formulas: since
holds for all
, thus
for all
.
Therefore, Theorem 1 (i) can be applied for Equation (
24) with
,
and
to obtain that every solution of Equation (
24) is oscillatory, which contradicts Inequality (
22). □
The following lemma may be helpful to verify the slowly varying property of A without having to evaluate it.
Lemma 2. For some and positive number κ, let be bounded and locally integrable, and be any function. If both p and τ are slowly varying at infinity, then so is the function To prove this lemma, we first need to state the following result (see Lemma 1.1 of [
17]).
Lemma 3. If is Lebesgue measurable and slowly varying at infinity, then, for all finite interval I, , as .
Proof of Lemma 2. For
, we have
From this and the triangle inequality, we obtain that, for any fixed
, the inequalities
hold. Now, if we let
, then the last two suprema vanish due to Lemma 3 and because
p is slowly varying. On the other hand, the last two terms also tend to 0, thanks to boundedness and to the slowly varying property of functions
and
p.
Therefore, holds for all . □
Note that, for A to be slowly varying, it is not sufficient to assume merely that at least one of p and is slowly varying. This is the case even under the additional assumptions of Theorem 3 on p and . This can be readily seen by considering examples and , and and , respectively. In both cases, function A will be -periodic, but nonconstant, so it cannot be slowly varying.
Our last theorem is a corollary of Lemma 2 and Theorem 3, and it gives another generalization of Theorem 1 of [
14] in case
p is bounded.
Theorem 4. For some positive numbers M and κ let and be continuous and slowly varying at infinity. Then Condition (12) implies that all solutions of Equation (1) are oscillatory. Proof. First, Lemma 2 infers that function
A from Equation (
11) is slowly varying. As already noted after Theorem 4 of [
15], the slowly varying property together with continuity implies uniform continuity. Hence,
p and
are uniformly continuous, so Theorem 3 applies, which finishes the proof. □
Let us briefly consider the case when
p is unbounded, and slowly varying. If we further assume that
holds for large
t, and
is such that there exists some
, for which
holds and
is nondecreasing (note that Theorem 1 of [
14] meets these assumptions), then, using the slowly varying property of
p, it can be easily shown that
. In particular, Condition (
4) is fulfilled, which yields that all solutions are oscillatory regardless of Condition (
12).
3. Example
Before concluding the paper, let us consider the following example, which may look a bit artificial. This is because our intention was to design it in such a way that—hopefully—no other known results could guarantee the oscillation of all solutions. Obviously, it is not possible to be aware of all the related results, and to check whether they are applicable; nevertheless, we shall exclude applicability of many classical, as well as many recent theorems.
Consider the equation
where
and
are small positive constants that will be determined later. Functions
p and
are clearly positive and bounded, so Equation (
26) is a special case of Equation (
1) with
Note that the functions
and
are slowly varying at infinity, since their derivatives vanish there (see Equation (
13)). This in turn yields that both
p and
are slowly varying, and, thus, in view of Lemma 2,
A is slowly varying as well.
On the other hand, a direct calculation shows that
Now, by setting
and
for all
, we obtain that
and
hold for all
. These together with Inequalities (
27) yield the estimates
and
Finally, for
, let
and
. Then, the above estimates take the form
It is now easy to see that, for all
, all assumptions of Theorem 3 (and also of Theorem 4) are fulfilled, and therefore all solutions are oscillatory. Note also that, since
as
, and
for all
, by choosing
small enough we can rule out the application of Conditions (
4), (
5) and various other sufficient conditions for the oscillation of all solutions of Equation (
26) (see e.g., conditions (C
)–(C
) from [
10]). Since function
is nonconstant, therefore neither Condition (
8) nor Theorem 2 can be applied to guarantee oscillation.