Note on Oscillation Conditions for First-order Delay Differential Equations

We consider explicit conditions for all solutions to linear scalar differential equations with several variable delays to be oscillatory. The considered conditions have the form of inequalities bounding the upper limit of the sum of integrals of coefficients over a subset of the real semiaxis, by the constant 1 from below. The main result is a new oscillation condition, which sharpens several known conditions of the kind. Some results are presented in the form of counterexamples.


Introduction
It follows from results by Ladas et al. [7] and Tramov [12] that all solutions of the equation ẋ(t) + a(t)x(t − τ) = 0, t ≥ 0, ( where a(t) ≥ 0 and τ = const > 0, are oscillatory in case lim sup t→+∞ t t−τ a(s) ds > 1.For an equation with variable delay, Corollary 2.1 from [7] presents the following oscillation condition.Suppose a ∈ C(R + , R + ), h ∈ C 1 (R + , R + ), h(t) ≤ t and h (t) ≥ 0 for all t ∈ R + , lim t→∞ h(t) = ∞, and lim sup t→∞ t h(t) a(s) ds > 1.Then all solutions of the equation ẋ(t) + a(t)x(h(t)) = 0, t ≥ 0, (1.2) are oscillatory.This result is extended and sharpened in many publications.In almost all of them the condition is imposed that the delay function h is nondecreasing.The present paper is devoted to conditions for all solution of the equation where a k (t) ≥ 0, h k (t) ≤ t, and h k (t) → ∞ as t → ∞, to be oscillatory.All new obtained oscillation conditions are generalizations of the results formulated above.We do not suppose that the functions h k are necessarily nondecreasing and accompany the obtained results by a number of counterexamples in order to compare the new oscillation conditions with known ones.
In Section 2 we discuss published results concerning oscillation conditions of the considered kind.In Section 3 our main result is obtained, and it is shown that known results are its corollaries.In Section 4 equation (1.2) is discussed.In Section 5 some ideas from the previous section are extended to the case of equation (1.3).Some results in the last three sections are represented in the form of counterexamples.

Known oscillation conditions
Theorem 2.1.3from the book [9] by Ladde et al. represents an oscillation condition for (1.2) that sharpens slightly the cited result from [7], as it is supposed that h ∈ C(R + , R + ), and the nonnegativity of h is replaced by the nondecrease of h.
This result is extended to the case of equation (1.3) in Theorem 3.4.3from the book [5] by Győri and Ladas.The basic oscillatory condition in the theorem is the inequality lim sup It is not stated explicitly that the functions h k are supposed to be nondecreasing, however, the authors did not mention anything to replace this condition.It is shown in Section 4 of this paper that the nondecrease is actually essential.
In [1, p. 36], there is an example showing that the inequality lim sup in contrast to that containing max in place of min, is not necessary for a nonoscillating solution to exist.In Section 3 of the present work we sharpen this result.
Tang [11] obtained an oscillation condition for the case of several constant delays which is not a consequence of the above conditions for (1.There are few published extensions of the considered oscillation conditions for the case of nondecreasing delay.The following result is by Tramov [12].If a(t) ≥ 0, t − h(t) ≥ h 0 > 0, lim t→∞ h(t) = ∞, and lim sup t→∞ t+h 0 t a(s) ds > 1, then every solution of (1.2) oscillates.In [12] the author also presented an example showing the sharpness of the constant 1: if it is diminished by arbitrary ε > 0, then the condition does not guarantee oscillation.
Koplatadze and Kvinikadze [6] obtained another oscillation condition for the case of nonmonotone delay.Suppose is sufficient for all solutions of (1.2) to be oscillatory.
Note that the nature of the considered oscillation conditions differs from that of the oscillation conditions of 1/e-type.This is expressed, in particular, in the possibility to extend the above oscillation condition to equations with oscillating coefficients.Such extension was apparently first made by Ladas at al. [8], their results sharpened by Fukagai and Kusano [4].Below we do not consider 1/e-type theorems and the problem of 'filling the gap' between 1/e and 1.A detailed discussion of this subject is found in the monographs [1][2][3] and the review [10].

Main result
Let parameters of equation (1.3) satisfy the following conditions for all k = 1, . . ., m: • the functions a k : R + → R are locally integrable; • the functions h k : R + → R are Lebesgue measurable; • a k (t) ≥ 0 and h k (t) ≤ t for all t ∈ R + .
We say that a locally absolutely continuous function x : R + → R is a solution to the equation if there exists a Borel initial function ϕ : (−∞, 0] → R such that the equality (1.3) takes place for almost all t ≥ 0, where x(ξ) = ϕ(ξ) for all ξ ≤ 0. Let us define a family of sets It follows from the stated above that all the sets of the family are Lebesgue measurable.
Proof.Suppose the conditions of the theorem are fulfilled and consider an arbitrary solution x of equation (1.3).Assume that x is not oscillatory.Without loss of generality, suppose that there exists t 0 ≥ 0 such that x(t) > 0 for all t ≥ t 0 .Then there exists t 1 ≥ t 0 such that h k (t) ≥ t 0 for all t ≥ t 1 and k = 1, . . ., m.It is obvious that x(t) is nonincreasing for all t ≥ t 1 .Further, there exists t 2 ≥ t 1 such that x(h k (t)) ≥ x(t) for all t ≥ t 2 and k = 1, . . ., m, and ∑ m k=1 E k (t 2 ) a k (s) ds > 1.There also exists t 3 > t 2 such that for all the sets which contradicts the assumption.Corollary 3.2.Suppose the functions h k are continuous and strictly increasing, lim t→∞ h k (t) = ∞ for k = 1, . . ., m, and Then every solution of equation (1.3) is oscillatory.
Proof.For each k = 1, . . ., m there exists the inverse function Then every solution of equation (1.3) is oscillatory.
Proof.We have
Proof.By virtue of the nondecrease of h k we have that The following example supplements Corollaries 3.2, 3.3 and 3.4.
On the other hand, lim sup It is obvious that Example 4.1 may be modified for the case that h is continuous.Consider Theorem 3.1 for the case m = 1.
The function h is not supposed to be nondecreasing in Corollary 4.2.The following corollaries represent an idea that to prove that all solutions to equation (1.2) are oscillatory it may be sufficient to consider an auxiliary equation with nondecreasing delay.In particular, this allows to establish oscillation in case the function h is not defined precisely.

Corollary 4.3. Let h
s) ds > 1 is derived from an oscillation condition obtained for an equation with distributed delay.It is shown in Section 3 that the above inequality cannot be replaced s) ds > 1.