Non-Linear Neutral Differential Equations with Damping: Oscillation of Solutions

: The oscillation of non-linear neutral equations contributes to many applications, such as torsional oscillations, which have been observed during earthquakes. These oscillations are generally caused by the asymmetry of the structures. The objective of this work is to establish new oscillation criteria for a class of nonlinear even-order differential equations with damping. We employ different approach based on using Riccati technique to reduce the main equation into a second order equation and then comparing with a second order equation whose oscillatory behavior is known. The new conditions complement several results in the literature. Furthermore, examining the validity of the proposed criteria has been demonstrated via particular examples.

Non-linear neutral differential equations have been extensively utilized to mathematically model several interesting phenomena that are observed in many areas of science and technology such as economics, biology, fluid dynamics, physics, differential geometry, engineering,control theory, materials science, and quantum mechanics. Asymptotic properties of solutions of non-linear neutral differential equations have been the objective of many researchers. Oscillation theory, however, has gained particular attention due to its widespread applications in mechanical oscillations, earthquake structures, clinical applications, frequency measurements and harmonic oscillators, which involve symmetrical properties; see the remarkable monograph of Hale [1].
Exploring the past few years, the asymptotic behavior of non-linear neutral differential equations has become a significant research area in different disciplines. In context of oscillation theory, it has been the object of research for many academics, who have investigated this notion for non-linear neutral differential and difference equations; the reader can refer to [2][3][4][5][6][7][8][9][10][11].
In [12], Liu et al. used the integral averaging technique to establish oscillation conditions for the solutions of the equation where L x = a(t) x (j−1) (t) On the other hand, in [13], the authors obtained oscillation criteria for equations with damping via comparing with first-order equations.
Continuing the investigation, the authors in [14,15] considered equation of the form and used the Riccati method to ensure that the equation is oscillatory if and lim inf t→∞ t γ(t) where Π(t) := γ j−1 (t)(1 − θ 1 (γ(t)))p 2 (t) The purpose of this paper is to improve and extend the results in [14,15] and establish new oscillation criteria for Equation (1). Our approach is based on the use of Riccati substitution to reduce Equation (1) into a second order equation and then compare it with a second order equation whose oscillatory behavior is known. For examining the validity of the proposed criteria, two examples with particular values are constructed.

Oscillation Conditions
The following lemmas are essential in the sequel.
Assume that x (j) (t) is of fixed sign and not identically zero on [t 0 , ∞) and that there exists a t 1 ≥ t 0 such that for every µ ∈ (0, 1).
Then, we have these cases: For the sake of simplification, we use some notations.
It is well known (see [20]) that if then (14) is oscillatory.

Conclusions
In this paper, we establish new oscillation criteria for a certain class of even-order non-linear differential equation with damping of the form (1). Our approach is different and obtained by using Riccati technique and comparing it with second-order equations.