Oscillatory Behavior of Second-Order Neutral Differential Equations

In this paper, we study oscillatory properties of neutral differential equations. Moreover, we discuss some examples that show the effectiveness and the feasibility of the main results.


Introduction
Delay differential equations are widely used in mathematical modeling to describe physical and biological systems, by inducing oscillatory behavior.
In the literature, several mathematical models with different levels of complexity have been proposed for delay differential equations in order to represent for example the cardiovascular system (CVS).
The pioneering and remarkable paper of Ottesen (1997) shows how to use delay differential equations to solve a cardiovascular model that has a discontinuous derivative. Ottesen (1997) also illustrated that complex dynamic interactions between nonlinear behaviors and delays associated with the autonomic-cardiac regulation may cause instability (Ataeea et al. 2015). Moreover, a model-based approach to stability analysis of autonomic-cardiac regulation was studied in Ataeea et al. (2015); specifically, it is important to underline that the autonomic-cardiac regulation operates by the interaction between autonomic nervous system (ANS) and cardiovascular system (CVS) (Ataeea et al. 2015).
It is clear that mathematical analysis related to physics-based models can be a versatile tool in examining delay differential equations from the point of view of medical and biological systems.
In this paper we consider the following equation of neutral type belonging to those families used to model problems that arise in the biological sciences. Our aim is to study the oscillatory behavior of (1.1) where w(y) = x(y)+ m 1 i=1 p i (y)x α i (ς i (y)), α i , γ and β j , for all i = 1, . . . , m 1 and j = 1, . . . , m 2 , are quotients of odd positive integers.
For a recent review on the asymptotic properties for functional differential equations (FDEs), we suggest to the reader the interesting book Berezansky et al. (2020).
and it satisfies the equation (1.1) for all y ∈ [y x , ∞).
We assume that (1.1) admits a solution in the sense of Definition 2.1.

Definition 2.2
A solution x(y) of (1.1) is said to be non-oscillatory if it is eventually positive or eventually negative; otherwise, it is said to be oscillatory.
Definition 2.3 Equation (1.1) is said to be oscillatory if all of its solutions are oscillatory.
In this paper, we restrict our attention to study oscillation and non-oscillation of (1.1). First of all, it is interesting to make a review in the context of functional differential equation. Brands (1978) proved that for each bounded delay ϑ(y), the equation is oscillatory if and only if the equation is oscillatory. Chatzarakis et al. (2019a) and Chatzarakis and Jadlovská (2019) considered the following more general equation and established new oscillation criteria for (2.1) when lim y→∞ A(y) = ∞ and lim y→∞ A(y) < ∞. Wong (2000) has obtained oscillation conditions of in which the neutral coefficient and delays are constants. In Baculíková and Džurina (2011a) and Džurina (2011), the authors studied the equation and established the oscillation of solutions of (2.2) using comparison techniques when γ = β = 1, 0 ≤ p(y) < ∞ and lim y→∞ A(y) = ∞. Using the same technique, Baculíková and Džurina (2011b) considered (2.2) and obtained oscillation conditions of (2.2) considering the assumptions 0 ≤ p(y) < ∞ and lim y→∞ A(y) = ∞. Tripathy et al. (2016), studied (2.2) and established several conditions of the solutions of (2.2) considering the assumptions lim y→∞ A(y) = ∞ and lim y→∞ A(y) < ∞ for different values of the neutral coefficient p. Bohner et al. (2017) obtained sufficient conditions for the oscillation of the solutions of (2.2) when γ = β, lim y→∞ A(y) < ∞ and 0 ≤ p(y) < 1. Grace et al. (2018) studied the oscillation of (2.2) when γ = β j , assuming that lim y→∞ A(y) < ∞, lim y→∞ A(y) = ∞ and 0 ≤ p(y) < 1. Li et al. (2019) established sufficient conditions for the oscillation of the solutions of (2.2), under the assumptions lim y→∞ A(y) < ∞ and p(y) ≥ 0. Karpuz and Santra (2019) studied the equation considering the assumptions lim y→∞ A(y) < ∞ and lim y→∞ A(y) = ∞, for different values of p.
For any positive, continuous and decreasing to zero function ρ : [y 0 , ∞) → R + , we set Let us assume that P(y) and U (y) are non-negative in [y 0 , ∞).
We now recall the technical lemmas and the main results contained in Bazighifan et al. (2020b).
hold for all y ≥ y 1 .

Oscillation Criteria for (1.1)
In this section we discuss our main results. The oscillation criteria in this paper complete the study started in Bazighifan et al. (2020b) but it is important to underline that the criteria discussed in Bazighifan et al. (2020b) differ from those examined in this work in terms of assumptions. Precisely, both the main results of Bazighifan et al. (2020b) (Theorem 1 and 2), require the existence of two constants δ 1 and δ 2 that are quotients of odd positive integers and the bounds for b j involve such constants. The results presented in this paper do not involve the existence of auxiliary constants and under fewer hypotheses guarantee the oscillatory behavior of the equations under consideration.
Proof Let the solution x be eventually positive. Then there exists y 0 > 0 such that x(y) > 0, x(ς i (y)) > 0 and x ϑ j (y) > 0 for all y ≥ y 0 and for all i = 1, 2, . . . , m 1 and i = 1, 2, . . . , m 2 . Applying Lemmas 2.1 and 2.3 for y ≥ y 1 > y 0 we conclude that w satisfies (2.3), w is increasing and x(y) ≥ P(y)w(y) for all y ≥ y 1 . From (1.1), we have for y ≥ y 1 . Applying (2.3) we conclude that lim y→∞ a(y) w (y) γ exists, and there exist y 2 > y 1 and a number c > 0 such that w(y) ≥ c for y ≥ y 2 . Integrating (3.1) from y 2 to y, for a suitable constantc, we havẽ which is a contradiction to (A6). The case where x is an eventually negative solution is similar and we omit it here. Thus, the proof is complete.
Remark Theorem 3.1 holds for any β j and γ .
Proof Proceeding as in the proof of Theorem 3.1 we obtain (3.1). Since w(y) is positive and increasing, ρ(y) is positive and decreasing to zero, there exists y 0 ≥ y 1 such that w(y) ≥ ρ(y) for y ≥ y 1 . (3.2) The rest of the proof is similar to that of Theorem 3.1 and hence it is omitted.
Proof Proceeding as in the proof of Theorem 3.1 we obtain (3.1). Now (3.1) can be written as for y ≥ y 2 > y 1 . Since w(y) A(y) is decreasing, there exists a constant k such that

w(y)
A(y) ≤ k for y ≥ y 2 . (3.5) Using (3.5) and β j < 1 in (3.4), we have The rest of the proof is similar to that of Theorem 3.2 and hence it is omitted.

Examples
We conclude the paper presenting some examples that show the effectiveness and the feasibility of the main results.

Conclusions
In this work we established several oscillation criteria for second-order nonlinear neutral differential equations. Our results complete the research started in Bazighifan et al. (2020b). For the sake of completeness, we presented some examples related to the main results of the paper.