Nonlinear Neutral Delay Differential Equations of Fourth-Order: Oscillation of Solutions

The objective of this paper is to study oscillation of fourth-order neutral differential equation. By using Riccati substitution and comparison technique, new oscillation conditions are obtained which insure that all solutions of the studied equation are oscillatory. Our results complement some known results for neutral differential equations. An illustrative example is included.


Definition 1.
A solution of (1) is said to be oscillatory if it has arbitrarily large zeros on [x y , ∞). Otherwise, a solution that is not oscillatory is said to be nonoscillatory. (1) is said to be oscillatory if every solution of it is oscillatory. Definition 3. A differential equation is said to be neutral if the highest-order derivative of the unknown function appears both with and without delay.

Definition 2. The Equation
Neutral differential equations are used in numerous applications in technology and natural science. For instance, the problems of oscillatory behavior of neutral differential equations have a number of practical applications in the study of distributed networks containing lossless transmission lines which arise in high-speed computers where the lossless transmission lines are used to interconnect switching circuits, see [1][2][3][4] . In fact, half-linear differential equations arise in a variety of real world problems such as in the study of p-Laplace equations non-Newtonian fluid theory, the turbulent flow of a polytrophic gas in a porous medium, and so forth; see [5][6][7]. During the past few years there has been interest by many researchers to study the oscillatory behavior of this type of equation, see [8][9][10][11][12]. Furthermore, many researchers investigate regularity and existence properties of solutions to difference equations; see [13][14][15] and the references therein. In [16], the authors studied oscillation conditions for equation where Φ = |s| p−2 s and n is even. The authors used Riccati substitution together with integral averaging technique.
In [17], Bazighifan obtained oscillation conditions for solutions of (1) and used comparison method with second-order equations. Moreover, in [16,18,19], the authors considered the equation where w(x) = y(x) + δ(x)y(g 2 (x)) and obtained a condition under which every solution of this equation is oscillatory. Bazighifan and Abdeljawad [20] give some results providing information on the asymptotic behavior of solutions of fourth-order advanced differential equations. This time, the authors used comparison method with first and second-order equations.
In this article, we establish oscillatory properties of solutions of (1). By using Riccati substitution and comparison technique, new oscillatory criteria for (1) are established. Our results complement some known results in literature. Furthermore, an illustrative example is provided.

Lemmas
The following lemmas will be used to establish our main results: Lemma 1. [21] Let β be a ratio of two odd numbers, D > 0 and G are constants. Then We consider the following notations: The following lemma summarizes the situations that are discussed in the proofs of our results.

Lemma 5.
[24] Let y be an eventually positive solution of (1). Then there exist two possible cases:

Main Results
Lemma 6. Let y be an eventually positive solution of (1). Then Proof. Let y be an eventually positive solution of (1). From the definition of w, we see that Repeating the same process, we obtain . Thus, (3) holds.
The second result of the paper is a theorem providing oscillation criterion for Equation (1). For this purpose, we employ the comparison method with first-order differential equations.
Institutional Review Board Statement: Not applicable.

Informed Consent Statement: Not applicable.
Data Availability Statement: Not applicable.