Symmetry and Its Importance in the Oscillation of Solutions of Differential Equations

: Oscillation and symmetry play an important role in many applications such as engineering, physics, medicine, and vibration in ﬂight. The purpose of this article is to explore the oscillation of fourth-order differential equations with delay arguments. New Kamenev-type oscillatory properties are established, which are based on a suitable Riccati method to reduce the main equation into a ﬁrst-order inequality. Our new results extend and simplify existing results in the previous studies. Examples are presented in order to clarify the main results.

It is clear that the form of problem Equation (1) is more general than all the problems considered in [12,14], where the authors in [12,14] discussed the oscillatory properties of differential equations of the neutral type with a canonical operator, and they used the comparison method and integral averaging technique to obtain these properties. Their approach is based on using these mentioned methods to reduce Equation (1) into a second-order equation , while in our article we discuss the oscillatory properties of differential equations with a middle term and with a non-canonical operator of the delay-type, and we employ a different approach based on using the Riccati technique to reduce the main equation into a first-order inequality to obtain more effective Kamenev-type oscillatory properties.
The aim of this article is to establish the oscillatory properties of Equation (1). Several studies have had very interesting results related to the oscillatory properties of solutions of differential equations.
Dzurina et al. [15] obtained sufficient conditions for oscillation for equation They also used the technique of comparison. In Grace et al. [16], some comparison criteria have been studied when τ(y) ≤ y, and some oscillation criteria for Equation (1) are given when Equation (2) holds.
In addition, the results obtained in [17] are presented for Equation (1) when and where there are positive functions A 1 , The purpose of this article is to explore the oscillation of Equation (1). New oscillation theorems are established, which are based on a suitable Riccati-type method.
This article is organized as follows. In Section 2, we introduce some auxiliary lemmas and some notations. In Section 3, we present new oscillation results for Equation (1) by Riccati transformation. Finally, two examples with specific values of parameters are offered to illustrate our main theorems.

Oscillation Criteria
In this section, we will give new oscillation criteria for Equation (1) by the Riccati technique. then In addition, if (N 2 ) holds and there exists a function where ζ(y) and w(y) are called Riccati transformations.
Proof. Let ξ be an eventually positive solution of Equation (1) hold. From Lemma 3 there exist two possible cases (N 1 ) and (N 2 ).

Theorem 1. Let Equation
and lim sup then Equation (1) is oscillatory.
Proof. Let ξ be a non-oscillatory solution of Equation (1). Without loss of generality, we can assume that ξ(y) is eventually positive. For case (N 1 ), from Lemma 4, we get that Equation (15) holds. Thus, we have Since y y 0 Thus, we get Hence, and so lim sup where a ∈ (0, 1) is a constant. Let j 1 = j 2 = j 3 = y, π(y) = ay, z(y) = y. Moreover, we have If we now set θ 1 (y) = θ 2 (y) = k = 1, we can easily find that the conditions of Theorem 1 are satisfied. So, Equation (52) is oscillatory. As a matter of fact, one such solution is ξ(y) = sin(y).

Conclusions
It's clear that the form of problem Equation (1) is more general than all the problems considered in [12,14]. In this paper, using the suitable Riccati-type transformation, we have offered some new sufficient conditions that ensure that any solution of Equation (1) oscillates under assumption ∞ y 0 1 j i (s) ds = ∞. In addition, it would be useful to extend our results to fourth-order differential equations of the form j 3 (y) j 2 (y) j 1 (y)ξ (y) β + m ∑ r=1 π r (y) f (ξ(z r (y))) = 0,