Even-order differential equation with continuous delay: nonexistence criteria of Kneser solutions

In this paper, we study even-order DEs where we deduce new conditions for nonexistence Kneser solutions for this type of DEs. Based on the nonexistence criteria of Kneser solutions, we establish the criteria for oscillation that take into account the effect of the delay argument, where to our knowledge all the previous results neglected the effect of the delay argument, so our results improve the previous results. The effectiveness of our new criteria is illustrated by examples.


Introduction
There is no doubt that the theory of oscillation of DEs is a fertile study area and has attracted the attention of many researchers recently. This is due to the existence of many important applications of this theory in various fields of applied science, see [18,19]. In the last decade, it is easy to notice the new research movement that aims to improve and develop the criteria for oscillations of DEs of different orders, see [3][4][5] and [9][10][11][12][13][14][15][16][17].
In detail, we consider the even-order delay DE of the form r · y (n-1) γ (ς) + A q · (y • g) γ ; a, b (ς) = 0, ς ≥ ς 0 , (1.1) Definition 1.1 A solution y of (1.1) is called a Kneser solution if there exists ς * ∈ I 0 such that y(ς)y (ς) < 0 for all ς ≥ ς * . (The set of all eventually positive Kneser solutions of (1.1) is denoted by K.) Definition 1.2 A solution y of (1.1) is said to be nonoscillatory if it is positive or negative, ultimately; otherwise, it is said to be oscillatory. The equation itself is termed oscillatory if all its solutions oscillate.
Next, let us briefly review a number of closely related results which motivated the present study.
In the paper, we are working on finding new criteria for oscillation of solutions of a class of even-order DEs in a noncanonical case. The paper is organized as follows. In Sect. 2, we present new conditions for the nonexistence of Kneser solutions of nonlinear evenorder DEs with continuous delay arguments. In Sect. 3, we are taking advantage of the new nonexistence criteria of Kneser solutions to create better criteria that ensure all solutions of (1.1) are oscillatory. In Sect. 4, we illustrate the effectiveness of our new criteria with examples. Now, we provide the lemmas that will be needed during the results.
Remark 2.1 Based on the definition of the class K, we note that y ∈ K if and only if y satisfies case (3).
Proof Assume that y ∈ K on [ς 1 , ∞). Integrating (1.1) from ς 1 to ς and using that fact that y (ς) < 0, we obtain for all ς ∈ I 1 . It follows from Lemma 2.2 that y converges to zero. Then there is ς 2 ∈ I 1 such that, for ς ≥ ς 2 , Next, by using the fact that (r 1/γ · y (n-1) ) ≤ 0, we see that Integrating (2.5) from ς to ∞ and taking the monotonicity of y (n-3) (ς) into account, we find Integrating again from ς to ∞, we obtain Going forward along the same method, we get for k = 0, 1, . . . , n -2. This completes the proof.
Proof Suppose to the contrary that y ∈ K on [ς 1 , ∞). From Lemma 2.3, we obtain (2.2) and (2.3) hold. Since g is delay w.s.t ς , we get y • g ≥ y for ς ≥ ς 2 and s ∈ [a, b]. Thus, (2.2) becomes Taking the limsup on both sides of the inequality, we arrive at contradiction with (2.6). This completes the proof.
For the next results, we introduce the following additional condition:

Lemma 2.4
Assume that y ∈ K, (2.1) hold and η is defined as in (2.6). Then there exists (2.7) Proof Assume that y ∈ K on I 1 . From Lemma 2.3, we obtain (2.2) and (2.3) hold. It follows from (2.2) and the fact that g(ς, s) ≤ ς that , which with (2.8) gives d dς Using this fact, one can easily see that This completes the proof.
Taking the limsup on both sides of the latter inequality, we obtain h η η ≤ 1. Then we obtain a contradiction with (2.9). This completes the proof.

Theorem 2.5 Assume that ( ) and (2.1) hold. If there exists a function
n-2 (ς) . (2.17) From (1.1), we see that Thus, from (2.7), (2.19) yields Therefore, from the definition of w, we get (2.20) Using inequality (1.2) with and ξ := w, we obtain , which, with (2.20), gives Integrating this inequality from ς 1 to ς , we arrive at . From (2.16), we are led to In view of (2.15), we get Taking the limsup, we obtain a contradiction. This completes the proof.

Oscillation criteria
In this section, we are taking advantage of new nonexistence criteria of Kneser solutions to create better criteria that ensure all solutions of (1.1) are oscillatory.