Criteria for the Nonexistence of Kneser Solutions of DDEs and Their Applications in Oscillation Theory

: The objective of this study was to improve existing oscillation criteria for delay differential equations (DDEs) of the fourth order by establishing new criteria for the nonexistence of so-called Kneser solutions. The new criteria are characterized by taking into account the effect of delay argument. All previous relevant results have neglected the effect of the delay argument, so our results substantially improve the well-known results reported in the literature. The effectiveness of our new criteria is illustrated via an example.

In this paper we are concerned with the study of the asymptotic behavior of the fourth-order delay differential equation: (a(l)(x (l)) α ) + f (l, x(τ(l))) = 0, l ≥ l 0 . (1) Throughout the paper, we assume α ∈ Q + odd := {β/γ : β, γ ∈ Z + are odd}, a ∈ C 1 (I 0 , R + ), a (l) ≥ 0, ∞ a −1/α ( )d < ∞, τ ∈ C(I 0 , R + ), τ(l) < l, lim l→∞ τ(l) = ∞, I := l , ∞ , If there exists a l x ≥ l 0 such that the real-valued function x is continuous, a(x ) α is continuously differentiable and satisfies (1), for all l ∈ I x , then x is said to be a solution of (1). We take into account these solutions x of (1) such that sup{|x(s)| : s ≥ l x } > 0 for every l x in I * . A solution x of (1) is said to be a Kneser solution if x(l)x (l) < 0 for all l ≥ l * , where l * is large enough. The set of all eventually positive Kneser solutions of Equation (1) is denoted by . A solution x of (1) is said to be non-oscillatory if it is positive or negative, ultimately; otherwise, it is said to be oscillatory. The equation itself is said to be oscillatory if all its solutions oscillate.
Below, we mention specifically some related works that were the motivation for this paper.
Zhang et al. [25] studied the oscillatory behavior of (1) when f (l, x) := q(l)x β . Results in [25] used an approach that leads to two independent conditions in comparison with first-order delay differential equations and a condition in a traditional form (lim sup(·) = +∞). However, to use (Lemma 2.2.3, [27]), they conditioned lim l→∞ x(l) = 0. Thus, under the conditions of (Theorem 1, [25]), Equation (1) still has a non-oscillatory solution that tends to zero. To surmount this problem, Zhang, et al. [38] considered-by using (Lemma 2.2.1, [27])-three possible cases for the derivatives of the solutions, and they followed the same approach as in (Theorem 1, [25]). However, in the case where x > 0, they ensured that lim l→∞ x(l) = 0, so they ensured that every solution of (1) is oscillatory.
By comparing with one or a couple of first-order delay differential equations, Baculikova et al. [14] studied the oscillatory behavior of (1) under the conditions In this study, we first create new criteria for the nonexistence of Kneser solutions of nonlinear fourth-order differential Equations (1). By using these new criteria, we introduce sufficient conditions for oscillation which take into account the effect of delay argument τ(l). All previous relevant results have neglected the effect of the delay argument, so our results substantially improve the well-known results reported in the literature. The effectiveness of our new criteria is illustrated via an example.
Proof. Suppose x ∈ . Integrating (1) from l 1 to l and using that fact that for all l ∈ I 1 . In view of (3), there is a l 2 ∈ I 1 such that Now, by using the monotonicity of a 1/α (l)x (l), we have Integrating (7) twice from l to ∞ and using and x(l) ≥ −a 1/α (l)x (l)δ 2 (l).
From (9) and (6), we see that Taking the limsup on both sides of the inequality, we arrive at (4). The proof is complete.

Lemma 3.
Assume that x ∈ and (2) hold. Then there exists a l ε ≥ l 1 such that d dl for any ε > 0 and l ∈ I ε . Moreover, if (H) holds, then Proof. Suppose x ∈ . Then, there is a l 1 ∈ I 0 such that x(τ(l)) > 0. Proceeding as in the proof of Lemma 2, we arrive at (6) and (8). Thus, for l ≥ l 2 , where l 2 ∈ I 1 is large enough, we have From the definition of η, for every ε > 0, there exists a l 3 ≥ l 2 such that for l ∈ I 3 . Hence, from (8), we have d dl , which with (6) gives d dl Using this fact, one can easily see that The proof is complete.

Lemma 4.
Assume that x ∈ and (H), (2) hold. Then Proof. Suppose x ∈ . Using Lemma 3, we get that (10) holds. As in the proof of Lemma 2, we have that (6) holds. From (6) and (10), we have Taking the limsup on both sides of the latter inequality, we obtain h η * η ≤ 1. Since ε is arbitrary, we obtain that (11) holds. The proof is complete.
From the previous results, the following theory can be inferred. Theorem 1. Assume that (2) holds. If one of the following conditions holds: then the set is empty.
Theorem 2. Assume (H) and (2) hold. If then the set is empty.
From (19), we are led to In view of (18), we get Taking limit supremum, we obtain a contradiction with (17). This completes the proof. h αη q(ζ)dζ > 1 (25) or lim sup then the set is empty.

Discussion and Applications
Depending on the new criteria for the nonexistence of Kneser solutions, we introduced new criteria for oscillation of (1). When checking the behavior of positive solutions of DDE (1), we have three Cases (1)- (3). In order to illustrate the importance of the results obtained for Case (3), we recall an existing criterion for a particular case of (1) with α = β: Theorem 3 (Theorem 2.1 with n = 4, [25]). Assume that α = β, and there exists a ρ ∈ C 1 (I 0 , R + ) such that lim sup l→∞ l l 0 for some λ ∈ (0, 1). Then every solution of (1) is oscillatory or tends to zero.
For Case (3), using Theorem 1, if one of the conditions (C 1 )-(C 3 ) holds, then we obtain a contradiction. The proof is complete.
For Case (3), using Corollary 1, if one of the conditions (25)- (27) holds, then we obtain a contradiction. The proof is complete.

Conclusions
We worked on extending and improving existing oscillation criteria for DDEs of the fourth order for the nonexistence of Kneser solutions. The new criteria that we proved are characterized by taking into account the effect of the delay argument.