On oscillation of second-order noncanonical neutral differential equations

In the present work, we study the second-order neutral differential equation and formulate new oscillation criteria for this equation. Our conditions differ from the earlier ones. Also, our results are expansions and generalizations of some previous results. Examples to illustrate the main results are included.

A solution of (1.1) is called oscillatory if it has arbitrarily many zeros on [l 0 , ∞), and is called nonoscillatory otherwise. Equation (1.1) is said to be oscillatory if all of its solutions are oscillatory.
Recently, the oscillatory theory of functional differential equations has received great attention due to the existence of a number of applications in engineering and the natural sciences. There are some contributions in the field of oscillatory behavior of different classes of differential equations, we refer the reader to [1][2][3][4][5][6][7][8][9][10][11][12] and the references mentioned therein. The neutral delay differential equations have applications in electrical networks containing lossless transmission lines; these networks appear in high-speed computers. See [13].
Some new oscillation criteria for the neutral nonlinear differential equation are established by Xu et al. [21], where α = 1, σ i (l) ≤ l and Agarwal et al. [22] investigated the second-order differential equations with a sublinear neutral term where 0 < α ≤ 1, λ(l) ≤ l and σ (l) ≤ l. They established some oscillation criteria under the condition (1.2) and (1.3). Dzurina [23] established a new comparison theorem for deducing oscillation of the nonlinear differential equation where α is a quotient of odd positive integers and σ (l) ≤ l. The objective of this paper is to study the oscillatory properties of the second-order neutral differential equations in noncanonical form. By using Riccati transformations, we present a new conditions for oscillation of the studied equation. The results obtained here extend and complement to some known results in the literature. See for example [21][22][23]. Some examples are provided to illustrate the relevance of new theorems.

Main results
In the rest of the work, we will adopt the following notation: In order to prove our results, we will present the following lemma.
for l ≥ l 1 .
Proof Let x(l) be a nonoscillatory solution of (1.1). Without loss of generality, we may assume that x(l) > 0, x(λ(l)) > 0 and x(σ (l)) > 0 for l ≥ l 1 ≥ l 0 . From (1.1), we have Hence, the function r(l)v (l) is decreasing and therefore we shall consider the following two cases, either v (l) < 0 or v (l) > 0. Assume that there exists and so which together with (2.4) implies that Define the function ω(l) by the Riccati substitution Then ω(l) > 0. Differentiating (2.7), using (1.1) and (2.5) we see that The above inequality, taking assumption (2.1) into account, implies that ω(l) → -∞ as l → ∞, which is a contradiction. Hence, the case v (l) > 0 is impossible. Thus, v(l) satisfies (2.2) for l ≥ l 1 . On the other hand, it follows from the monotonicity of From (2.11) and (2.12), we conclude that The proof of the lemma is complete. (2.14) then every solution x(l) of (1.1) is oscillatory.
Integrating (2.17) from l 1 to l, we obtain The above inequality, taking assumptions (2.13) and (2.14) into account, implies that v(l) → -∞ as l → ∞, which is a contradiction. The proof of the theorem is complete.
Proof Let x(l) be a nonoscillatory solution of (1.1). Without loss of generality, we may assume that x(l) > 0, x(λ(l)) > 0 and x(σ (l)) > 0 for l ≥ l 1 ≥ l 0 . Because of (2.1), from Lemma 2.1, we can conclude that v(l) satisfies (2.2). Now, since Hence, we observe that (l) = v(l) + r(l)v (l)π(l) ≥ 0 is nonincreasing. By integrating (2.22) from l to ∞ and using (2.11), we obtain Using (2.23) and (2.24), we see that W (l) = -r(l)v (l) is a positive solution of the differential inequality From the increasing property of W (l), we get that is, which is a contradiction. The proof of the theorem is complete.
then every solution x(l) of (1.1) is oscillatory.
(2.30) From (2.10) and (2.26), we have In view of (2.30) and (2.31), we obtain which is a contradiction. The proof of the theorem is complete. Now, we will present examples to illustrate our main results.

Conclusions
In this work, we have obtained some new oscillation criteria for (1.1) in the case where v(l) := x α (l) + q(l)x(λ(l)). These results ensure that all solutions of the equation studied are oscillatory. The results obtained here extend and complement some known results in the literature. See for example [21][22][23]. It will be of interest to investigate the higher-order equations of the form r(l) x α (l) + q(l)x λ(l) where g(x) ≥ kx α .