Asymptotic Properties of Neutral Differential Equations with Variable Coefficients

The aim of this work is to study oscillatory behavior of solutions for even-order neutral nonlinear differential equations. By using the Riccati substitution, a new oscillation conditions is obtained which insures that all solutions to the studied equation are oscillatory. The obtained results complement the well-known oscillation results present in the literature. Some example are illustrated to show the applicability of the obtained results.


Introduction
Neutral differential equations appear in models concerning biological, physical and chemical phenomena, optimization, mathematics of networks, dynamical systems and their application in concerning materials and energy as well as problems of deformation of structures, elasticity or soil settlement, see [1].

Definition 2.
A solution of (1) is called oscillatory if it has arbitrarily large zeros on [ς x , ∞), and otherwise is called to be nonoscillatory. (1) is said to be oscillatory if all its solutions are oscillatory.

Definition 3. The Equation
We collect some relevant facts and auxiliary results from the existing literature. Bazighifan [2] using the Riccati transformation together with comparison method with second order equations, focuses on the oscillation of equations of the form where n is even. Moaaz et al. [27] gives us some results providing informations on the asymptotic behavior of (1). This time, the authors used comparison method with first-order equations.
To prove our main results we need the following lemmas: 28]). Let α ≥ 1 be a ratio of two odd numbers. Then for all sufficient large ς.
Assume that x (n) (ς) is of fixed sign and not identically zero on [ς 0 , ∞) and that there exists a ς 1 is eventually of one sign for all large ς, then there exists a ς x > ς 1 for some ς 1 > ς 0 and an integer m, 0 ≤ m ≤ n with n + m even for

Remark 1.
It is easy to verify that B [·; l, ς] is a linear operator and that it satisfies Lemma 5. Assume that x is an eventually positive solution of (1) and Proof. Let x be an eventually positive solution of (1) on [ς 0 , ∞). From (1), we see that Thus, for all sufficiently large ς, we have From (8) and the definition of z, we get Thus, by using (8) and (9), we obtain From (5), we get This completes the proof.

Lemma 8.
Let (2) holds and assume that x is an eventually positive solution of (1). If there exists positive for some µ ∈ (0, 1), then z not satisfies case (I 2 ) .