New monotonicity properties and oscillation of n -order functional differential equations with deviating argument

. In this paper, we offer new technique for investigation of the even order linear differential equations of the form


Introduction
We consider the general higher order differential equation with deviating argument y (n) (t) = p(t)y(τ(t)). (E) Throughout the paper, it is assumed that n is even and the following conditions hold (H 1 ) p(t) ∈ C 1 ([t 0 , ∞)), p(t) > 0, By a proper solution of Eq. (E) we mean a function y : [T y , ∞) → R which satisfies (E) for all sufficiently large t and sup{|y(t)| : t ≥ T} > 0 for all T ≥ T y . We make the standing hypothesis that (E) does possess proper solutions.
As is customary, a proper solution y(t) of (E) is said to be oscillatory if it has arbitrarily large zeros. Otherwise, it is said to be nonoscillatory. The equation itself is termed oscillatory if all its proper solutions oscillate.
Oscillation phenomena appear in different models from real world applications; see, for instance, the papers [15][16][17] for models from mathematical biology where oscillation and/or delay actions may be formulated by means of cross-diffusion terms. The problem of establishing oscillation criteria for differential equations with deviating arguments has been a very active research area over the past decades (see [1]- [18]) and several references and reviews of known results can be found in the monographs by Agarwal et al. [1], Došlý and Rehák [5] and Ladde et al. [18].
It is known that the set N of all nonoscillatory solutions of (E) has the following decom- where y(t) ∈ N ℓ means that there exists t 0 ≥ T y such that Such a y(t) is said to be a solution of degree ℓ. Following Kiguradze [7], we say that equation The reason for such definition is the observation that (E) with τ(t) ≡ t always possesses solutions of degrees 0 and n, that is N 0 ̸ = ∅ and N n ̸ = ∅. The situation when τ(t) ̸ ≡ t is different. In fact, it may happen that N 0 = ∅ or N n = ∅ when the deviation |t − τ(t)| is sufficiently large. This remarkable fact was first observed by Ladas et al. [18]. Later Koplatadze and Chanturia [11] have shown that (E) does not allow solutions of degree 0 if τ(t) ≤ t and and (E) does not allow solutions of degree n provided that τ(t) ≥ t and lim sup t→∞ On the other hand, Koplatadze et al. [12] proved that (E) enjoys property (B) if τ(t) ≤ t and lim sup s(τ(s)) n−2 p(s) ds In this paper, we establish new technique that essentially improves (1.2) and (1.3), which leads to qualitative better criteria for bounded or unbounded oscillation of (E). Our approach essentially involves establishing stronger monotonicities for the positive solutions of (E) than those presented in known works.

Main results
Now we are introduce new monotonicity for nonoscillatory solution y(t) ∈ N 0 of (E).
Proof. Assume on the contrary that y(t) is an eventually positive solution of (E) such that y(t) ∈ N 0 , and lim t→∞ y(t) = ℓ > 0. Then y(τ(t)) > ℓ, eventually, let us say for Integrating again from t to ∞ and changing the order of integration, we have Repeating this procedure, we are led to where the last integration was from t 1 to ∞. Condition (2.2) contradicts (2.1) and we conclude that y(t) → 0 as t → ∞.
Proof. Assume that y(t) ∈ N 0 is an eventually positive solution of (E). It follows from (E) that In view of Corollary 2.2, an integration of (E) from t to ∞ yields Integrating again from t to ∞ and changing the order of integration, we have Repeated reusing of this procedure yields Setting (2.4) into (2.3) and taking (E) into account, one gets Therefore and we conclude that e β 0 (t) y (n) (t) is decreasing, which is, in view of (E), equivalent to the fact that p(t)y(τ(t))e β 0 (t) is decreasing.
Employing the above-mentioned monotonicity we are prepared to present criterion for bounded oscillation of (E). Proof. We argue by contradiction. Assume that (E) possesses an eventually positive solution y(t) ∈ N 0 . Integrating (E) from u to t (u ≤ t) and using the monotonicity of p(t)y(τ(t))e β 0 (t) , we have −y (n−1) (u) ≥ t u p(s)y(τ(s)) ds ≥ y(τ(t))p(t)e β 0 (t) t u e −β 0 (s) ds.
Integrating the above inequality from u to t and changing the order of integration leads to Proceeding in the same way (n − 2)-times, we finally get which is contraction with (2.5) and we conclude, that class N 0 is empty. Moreover, thanks to (1.4) every nonoscillatory solution of (E) is of degree n.
Example 2.5. Consider the delay differential equation It is easy to see that (1.4) holds true. Since α 0 (t) = p 0 (n−1)! τ n−1 = ω and β 0 (t) = ωt, condition Then where k is a positive integer such that (2.8) holds, then a bounded oscillatory solution of (E) is y(t) = sin( n √ p 0 )t. Example 2.6. We consider the delayed Euler differential equation It is easy to see that (1.4) reduces to On the other hand, Using notation condition (2.5) takes the form which guarantees that N 0 = ∅ for (E x2 ). If in addition (2.9) holds, then every nonoscillatory solution of (E x2 ) is of degree n. For n = 2 (n = 4) and λ = 0.5 condition (2.10) is satisfied when while (1.2) requires p 0 > 5.1774 (p 0 > 226.58). So our progress is significant. Now we turn our attention to bounded oscillation of (E). We set Theorem 2.7. Let y(t) ∈ N n , τ(t) ≥ t. Then |y(τ(t))|p(t)e −β n (t) is increasing.
Proof. Assume that y(t) ∈ N n is an eventually positive solution of (E). An integration of (E) from t 1 to t yields p(s)y(τ(s)) ds.
Integrating the last inequality from t 1 to t and and changing the order of integration, we obtain Repeating this procedure, we have Consequently, where we have used that y(τ(t)) is increasing. By combining inequalities (2.3) and (2.11), we conclude that which in view of (E) implies Consequently, and we conclude that e −β n (t) y (n) (t) is increasing, which is in view of (E) means that p(t)y(τ(t))e −β n (t) is increasing function. The proof is completed.
We use the above-mentioned monotonicity to establish criterion for unbounded oscillation of (E). Proof. Assume on the contrary that (E) possesses an eventually positive solution y(t) ∈ N n . Integrating (E) from t to u (t ≤ u) and using the monotonicity of p(t)y(τ(t))e −β n (t) , we have y (n−1) (u) ≥ y(τ(t))p(t)e −β n (t) u t e β n (s) ds.
Integrating again from t to u and changing order of integration, we get (2.13) Proceeding in the same way (n − 2)-times, we finally obtain This contradiction establishes the desired result and the proof is completed.
Proceeding exactly as in Example 2.5 we are led to (2.8) which by Theorem 2.8 ensures that every nonoscillatory solution of (E x3 ) is of degree 0 or in other words, every unbounded solution (if exists) of (E x3 ) is oscillatory.
On the other hand, as Remark 2.11. In this paper, we have introduce new technique for investigation of monotonicity for nonoscillatory solutions of higher order differential equations. The monotonicities obtained have been applied to establish new criteria for all solutions to be of degree 0 or to be of degree n.