Neutral Differential Equations of Fourth-Order: New Asymptotic Properties of Solutions

: In this work, we will derive new asymptotic properties of the positive solutions of the fourth-order neutral differential equation with the non-canonical factor. We follow an improved approach that enables us to create oscillation criteria of an iterative nature that can be applied more than once to test oscillation. In light of this, we will use these properties to obtain new criteria for the oscillation of the solutions of the studied equation. An example is given to show the applicability of the main results

We say that a real-valued function u ∈ C 3 (I u ), u ≥ 0 , is a solution of (1) if r(φ ) ∈ C 1 (I u ), u satisfies (1) on I u , and sup{|u( )| : 1 ≤ 0 } > 0 for every 1 ∈ I u .A solution of (1) is called oscillatory if it has arbitrarily large zeros on I u ; otherwise, it is called nonoscillatory.The equation itself is called oscillatory if all its solutions oscillate.

Remark 1.
As usual, all occurring functional inequalities are assumed to hold finally, that is, they are satisfied for all large enough.Moreover, we evaluate some integrals on the extended real line.
Fourth-order differential equations are quite often encountered in mathematical models of various physical, biological, and chemical phenomena.Applications include, for in-stance, problems of elasticity, deformation of structures, or soil settlement; see [1].Questions related to the existence of oscillatory and nonoscillatory solutions play an important role in mechanical and engineering problems.In natural science and technology, neutral differential equations have a wide range of applications.They are often employed, for example, in the study of distributed networks with lossless transmission lines (see Hale [2]).
In the following, we show some previous results in the literature.Many authors in [7][8][9] studied the asymptotic properties of the solutions of equation where r ( ) > 0 and In [10,11], Zhang et al. studied the oscillation of (3) under the assumption that El-Nabulsi et al. [12] investigated the oscillation properties of solutions to the fourthorder nonlinear differential equations r( ) u ( ) where f (φ)/φ α ≥ k > 0, for φ = 0 and (4) holds Zhang et al. [13] and Moaaz et al. [14] studied the oscillation of (6) under the condition (5).By using the technique of comparison with first order delay equations, Xing et al. [15] established some oscillation theorems for equation under the condition (4).Chatzarakis et al. [16] established some oscillation criteria for neutral differential equation under the assumption (4).Very recently, By using Riccati transform, Dassios and Bazighifan [17] proved that the fourth-order nonlinear differential equation r( ) φ ( ) is almost oscillatory, under the condition (5).
On the other hand, there are other techniques for studying the oscillatory behavior of differential equations by analysis of the characteristic equation and its roots.For example very recently Pedro in [18] obtains sufficient conditions under which the system has at least a nonoscillatory solution, based on the form of the system matrices, are obtained via the analysis of the characteristic equation.Also in [19] the numerical controllability of an integro-differential equation is briefly discussed.Additionally, the authors consider as keytools: the Laplace transform, the Mittag-Leffler matrix function and the iterative scheme.
Our results here is based on creating new comparison theorems that compare the 4th-order equation with first-order delay differential equations.We establish new oscil-lation criteria for a class of fourth-order neutral differential equations.This new results improves a number of results reported in the literature.Example is provided to illustrate the main results.Lemma 1 ([5]).Assume that w ∈ C n (I 0 , R + ), w (n) is of one sign, eventually.Then, there exist a κ w ∈ I 0 and κ ∈ [0, n] is integer, with (−1) n+κ w (n) (κ) ≥ 0 such that for all κ ∈ I w .
Lemma 3. Assume that u is a positive solution of (1) and φ satisfies case (C),then Proof.Assume that u is a positive solution of (1) and φ satisfies case (C).From ( 1) and (H 2 ), we have that r( )φ ( ) is nonincreasing, and hence Since φ ( ) is a positive decreasing function, we have that φ ( ) converges to a nonnegative constant when → ∞.Thus, (8) becomes from ( 9), we get φ ( ) This implies φ ( ) This implies Thus, the proof is complete.
(B 1 ): Using case (C), we have that Integrating this relationship twice over [ , ∞), and taking into account the behavior of derivatives in case (C), we arrive at (B 1 ).
has a positive solution, where δ j and λ are defined as in Theorem 2.
Proof.Assume that u is a positive solution of (1).Then from Lemma 2, we get the cases (A)-(C).In view of [21], the fact that the solutions of Equation ( 23) oscillate and the condition (24) is fulfilled, rules out the cases (A) and (B), respectively.Then, we have (C) hold.Using Theorem 3, we get that Equation ( 19) has a positive solution, a contradiction.Therefore, the proof is complete.
Remark 2. By reviewing the results in [23] and by choosing η 0 = 2 and p 0 = 0.12 we have that (27) is oscillatory if h 0 > 2. 206 3. It is easy to note that this condition essentially neglects the influence of delay argument θ(t).However, our criterion (28) takes into account the influence of θ 0 .Furthermore, using (28) and m = 0, every solution of the differential equation: is oscillatory, despite the failure of the results in [23].
Remark 4. In the non-neutral case, that is, p 0 = 0, the oscillation condition of the Equation ( 27) becomes: If m = 0 and θ 0 = 1.103 66, then we have the same condition obtained in [6,10].

Conclusions
In this work, we studied the oscillatory behavior of a class of fourth-order neutral differential equations were presented.In the noncanonical case, we obtained new criteria based on comparison principles that ensure the oscillation of all solutions of the studied equation.By comparing with previous results, we found that our results are easy to apply and do not require unknown functions.Moreover, the new criteria have an iterative nature.An interesting problem is to extend our results to even-order neutral differential equations.