Oscillation and Asymptotic Properties of Differential Equations of Third-Order

: The main purpose of this study is aimed at developing new criteria of the iterative nature to test the asymptotic and oscillation of nonlinear neutral delay differential equations of third order with noncanonical operator ( a ( where ι ≥ ι 0 and w ( ι ) : = x ( ι ) + p ( ι ) x ( ι − τ ) . New oscillation results are established by using the generalized Riccati technique under the assumption of (cid:82) → . Our new results complement the related contributions to the subject. An example is given to prove the signiﬁcance of new theorem.


Introduction
The objective of this paper is to provide oscillation theorems for the third order equation as follows: where a(ι), b(ι), p(ι), q(ι) ∈ C([ι 0 , +∞)), a(ι), b(ι) > 0, a (ι) ≥ 0, q(ι) ≥ 0, β ≥ 1 and 0 ≤ p(ι) ≤ p 0 ≤ 1. The main results are obtained under the following assumptions: We intend that for a solution of (1), we mean a function x(ι) ∈ C([T x , ∞)), T x ≥ ι 0 , which has the property w ∈ C 1 ([T x , ∞)), b w ∈ C 1 ([T x , ∞)), a ((b w ) ) β ∈ C 1 ([T x , ∞)) and satisfies (1) on [T x , ∞). We only consider those solutions x of (1) which satisfy sup{|x(ι)| : ι ≥ T} > 0 for all T ≥ T x . We start with the assumption that Equation (1) does possess a proper solution. A proper solution of Equation (1) is called oscillatory if it has a sequence of large zeros lending to ∞; otherwise, we call it non-oscillatory. Neutral/delay differential equations of the third order are used in a variety of problems in economics, biology, and physics, including lossless transmission lines, vibrating masses attached to an elastic bar, and as the Euler equation in some variational problems; see Hale [1]. As a result, there is an ongoing interest in obtaining several sufficient conditions for the oscillation or non-oscillation of the solutions of different kinds of differential equations; see  as examples of instant results on this topic.
However, to the best of our knowledge, only a few papers have studied the oscillation of nonlinear neutral delay differential equations of third order with distributed deviating arguments; see, for example, [2][3][4][5]. Recently, Haifei Xiang [6] and Haixia Wang et. al [7] studied the oscillatory behavior of Equation (1) under the following assumption: Motivated by this above observation, in this paper, we extend the results under the following assumption: Motivated by these reasons mentioned above, in this paper, we extend the results using generalized Riccati transformation and the integral averaging technique. We establish criteria for Equation (1) to be oscillatory or converge to zero asymptotically with the assumption of (2). As is customary, all observed functional inequalities are assumed to support eventually; that is, they are satisfied for all ι that are large enough.

Main Results
For our further reference, let us denote the following: and and Then, every solution x(ι) of (1) is either oscillatory or tends to 0.
We will present an example to illustrate the main results.

A Concluding Remark
We established new oscillation theorems for (1) in this paper. The main outcomes are proved via the means of the integral averaging condition, and the generalized Riccati technique under the assumptions of ι ι 0 a −1/β (s)ds < ι ι 0 1 b(s) ds = ∞ as ι → ∞. Examples are given to prove the significance of the new results. The main results in this paper are presented in an essentially new form and of a high degree of generality. For future consideration, it will be of great importance to study the oscillation of (1) when −∞ < p(ι) ≤ −1 and |p(ι)| < ∞.