Oscillation Criteria for First Order Differential Equations with Non-Monotone Delays

: New sufﬁcient criteria are obtained for the oscillation of a non-autonomous ﬁrst order differential equation with non-monotone delays. Both recursive and lower-upper limit types criteria are given. The obtained results improve most recent published results. An example is given to illustrate the applicability and strength of our results.


Introduction
Consider the first order delay differential equation where p, τ ∈ C([t 0 , ∞), [0, ∞)) and τ(t) < t for t ≥ t 0 , such that lim t→∞ τ(t) = ∞. A solution of Equation (1) is a function x(t) on [t, ∞), wheret = min t≥t 0 τ(t), which is continuously differentiable on [t 0 , ∞) and satisfies Equation (1) for all t ≥ t 0 . As customary, a solution of Equation (1) is called oscillatory if it has arbitrarily large zeros. Equation (1) is said to be oscillatory if all its solutions are oscillatory.
In 1972, Ladas et al. [13] proved that Equation (1) where the delay τ(t) is assumed to be a nondecreasing function.
In 1979, Ladas [12] (for Equation (1) with constant delay) and in 1982, Koplatadze and Chanturija [10] established the celebrated oscillation criterion The oscillation of Equation (1) has been studied when 0 < k ≤ 1 e , L ≤ 1 and τ(t) is nondecreasing, see [8,9,15,16] and the references cited therein. In most of these works, the oscillation criteria have been formulated as relations between L and k. For example, Jaros and Stavroulakis [8], Kon et al. [9], Philos and Sficas [15], and Sficas and Stavroulakis [16] obtained the following criteria, respectively: where λ(k) is the smaller real root of the equation λ = e λk . The same problem has been considered for Equation (1) with non-monotone delays, see [2,4,11,[17][18][19]. The latter case is much more complicated than the monotone delays case. In fact, according to Braverman and Karpuz ([2], Theorem 1), condition (2) does not need to be sufficient for the oscillation of Equation (1) if τ(t) is non-monotone. To overcome this difficulty, many authors used a nondecreasing function δ(t) defined by: hence, many results were obtained by using techniques similar to those of the monotonic delays case. Most of these results were given by recursive formulas. Next, we give an overview of such results: In 1994, Koplatadze and Kvinikadze [11] proved the following interesting result which requires the definition of the sequence of functions {ψ i } ∞ i=1 as follows: where k, δ, and ψ j , are defined respectively by (3), (5), and (6) and Then, Equation (1) is oscillatory.
In 2011, Braverman and Karpuz [2] obtained the following sufficient condition for the oscillation of Equation (1), In 2014, Stavroulakis [17] improved condition (8) to In 2015, Infante et al. [19] proved that Equation (1) is oscillatory if one of the following conditions is satisfied: where g(t) is a nondecreasing function satisfying that τ(t) ≤ g(t) ≤ t for all t ≥ t 1 and some t 1 ≥ t 0 .
In 2016, El-Morshedy and Attia [4] proved that Equation (1) is oscillatory if there exists a positive integer n such that where k * := lim inf t→∞ t g(t) p(s) ds, c, g are defined as before, and {q n (t)} is given by q n−1 (s)e t g(s) q n−1 (u)du ds, n = 2, 3, . . . .
Very recently, Bereketoglu et al. [18] proved that Equation (1) oscillates if for some ∈ N the following criterion holds where In this work, we obtain new sufficient criteria of recursive type for the oscillation of Equation (1), when the delay is non-monotone and k * ≤ 1 e <L < 1, whereL := lim sup t→∞ t g(t) p(s)ds. In addition, new practical lower limit-upper limit type criteria similar to those in [8,9,15,16] are obtained. These new conditions improve some results in [2,5,8,9,11,13,[16][17][18][19]. An illustrative example is given to show the strength and applicability of our results.

Main Results
Throughout this work, we assume that c, g, k * , λ, t 1 are defined as above and g i (t) stands for the ith composition of g.

Lemma 1.
Assume that x(t) is an eventually positive solution of Equation (1). Then, for all sufficiently large t. (1), there exists a sufficiently large T > t 1 such that x(t) satisfies eventually

Proof. Since x(t) is an eventually positive solution of Equation
Using ( [5], Lemma 2.1.2), for sufficiently small > 0 and sufficiently large t, we have On the other hand, dividing both sides of Equation (1) by x(t) and integrating the resulting equation from s to t, s ≤ t, we obtain Therefore, Integrating Equation (1) from τ(ξ) to g(ξ), Using (14) as well as the nonincreasing nature of x(t), it follows that Thus, This together with (16) gives Since (15) implies that x(s) ds , (17) yields Repeating this process, we arrive at the following inequality On the other hand, by integrating Equation (1) from τ(t) to t, we have Using (15), we obtain x(τ(s)) = x(t)e t τ(s) p(u) x(τ(u)) x(u) du . Therefore, (19) implies that Now, substituting (18) into (20), we have From the last inequality and (15), we obtain It follows from this and (15) that A simple induction implies that Substituting the previous inequality into (20), we get Therefore, by using the same arguments, as before, we obtain Theorem 2. Assume that k * ≤ 1 e and m, n ∈ N such that Then, every solution of Equation (1) is oscillatory.
Proof. Assume the contrary, i.e., there exists a non-oscillatory solution x(t). Due to the linearity of Equation (1), one can assume that x(t) is eventually positive. Now, integrating Equation (1) we obtain By using (15), it follows that Therefore, Lemma 1 yields Substituting into (22), we get for sufficiently large t. Therefore, .
The proofs of the following two results are basically similar to that of Lemma 1 and Theorem 2. p(s)e where ∈ (0, λ(k * )). Then, all solutions of Equation (1) oscillate. , and w(t) := x(g(t)) x(t) .
The proof of the following theorem is a consequence of Lemmas 1, 2, and ( and If one of the following conditions is satisfied: then every solution of Equation (1) is oscillatory. (1) has a nonoscillatory solution x(t); as usual, we assume that x(t) is an eventually positive solution. Let
To prove case (ii), integrating Equation (1) from g 2 (t) to g(t), we obtain which, by using the nonincreasing nature of x(t) and the assumption that τ(t) ≤ g(t), implies that x(g(t)) − x(g 2 (t)) + x(g 2 (t)) In view of (27), we have p(s)ds.
Substituting into (33), it follows that From this and (29), we obtain Again Lemma 2 and (32) imply for sufficiently small that t g(t) p(s)ds + 1 However, as in the proof of case (i), we have t g(t) p(s) Combining the inequalities (34) and (35), we obtain Letting → 0, we obtaiñ
(ii) It is easy to show that the conclusion of Theorem 6 is valid, if p(t) > 0 and condition (27) is replaced by lim inf t→∞ p(g(t))g (t) p(t) = 1.
The following example illustrates the applicability and strength of our result.
Author Contributions: All authors contributed equally to the research and to writing the paper. All authors have read and agreed to the published version of the manuscript. .

Funding:
This research received no external funding.