New oscillation theorems for a class of even-order neutral delay differential equations

In this work, we study the oscillatory behavior of even-order neutral delay differential equations υn(l)+b(l)u(η(l))=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\upsilon ^{n}(l)+b(l)u(\eta (l))=0$\end{document}, where l≥l0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$l\geq l_{0}$\end{document}, n≥4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$n\geq 4$\end{document} is an even integer and υ=u+a(u∘μ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\upsilon =u+a ( u\circ \mu ) $\end{document}. By deducing a new iterative relationship between the solution and the corresponding function, new oscillation criteria are established that improve those reported in (T. Li, Yu.V. Rogovchenko in Appl. Math. Lett. 61:35–41, 2016).

As is customary, a solution of Eq. (1.1) is said to be oscillatory if it is neither eventually positive nor eventually negative on [t 0 , ∞) and otherwise, it is termed nonoscillatory.
The importance of studying neutral delay differential equations comes from their emergence when modeling many phenomena in different applied sciences, see [2,3]. The qualitative theory of various classes of neutral differential equations has become an important area of research due to the fact that such equations arise in a variety of real world prob-lems such as in the study of non-Newtonian fluid theory and porous medium problems; see [4].
In 2016, Li and Rogovchenko [1] studied the oscillatory behavior of solutions of neutral delay equation (1.1). They used an approach similar to that used in [18], and established the relationship between u and υ to have the form By using the comparison with the first-order delay equations, they obtained improved criteria over the previous ones in the literature. In this paper, by improving the relationship (1.2), we establish a new criterion that improves the results in [1]. An example is given to illustrate the importance of our results.
In order to discuss our main results, we need the following auxiliary lemmas.

Lemma 2.1
Assume that u is an eventually positive solution. Then, we have two cases for the derivatives of υ as Based on the facts that n is even and υ n (t) ≤ 0, cases (1) and (2) are deduced directly from Lemma 1.3.

Theorem 2.1 Assume that μ (t) > 0 and there exists an even integer m such that
for all k = 1, 2, . . . , n/2. Suppose that there exist functions and If there exist λ i ∈ (0, 1), i = 0, 1, 2, such that the first-order delay equations and are oscillatory, every solution of Eq. (1.1) is oscillatory.
Proof Assume that Eq. (1.1) has an eventually positive solution u. It follows from (1.1) that υ n (t) = -b(t)u(η(t)) ≤ 0. Thus, using Lemma 2.1, we see that there are two cases for the derivatives of υ for large t, either (1) or (2).
Assume that (1) holds. Since υ is an increasing positive function, we obtain lim t→∞ υ(t) = 0. Therefore, by virtue of Lemma 1.1, we get for every λ ∈ (0, 1) and for all large t. It follows from the definition of υ(t) that and ) .
Remark 2.4 Although the results of Li and Rogovchenko in [1] improved their previous results, they used Lemma 1.2 with λ = 1 (and this is inaccurate); see Remark 12 in [20]. Theorem 2.1, with n = 2, is a correction of Theorem 2.1 in [1]. Moreover, our results improve the results in [1], since the iterative nature of the two functions a(t) and a(t) enables us to test for oscillations, even when the previously known results fail to apply. Let us consider a special case of (2.31), namely, u(t) + 10u(0.9t) (4) + 110 t 4 u(0.5t) = 0. (2.33) We note that the condition (2.32) fail to apply on (2.33) when n = 2, 4 (consequently, the results in [1] also fail). But, at n = 6, the condition (2.32) is satisfied. Therefore, our results improve the previous results in the literature.
Remark 2.5 It would be of interest to further investigate Eq. (1.1) with different neutral coefficients; see [27] and [28] for more details. It would also be interesting to extend this development to higher-order nonlinear neutral differential equations.