Oscillatory criteria of noncanonical even-order differential equations with a superlinear neutral term

The oscillatory behavior of solutions of an even-order diﬀerential equation with a superlinear neutral term is considered using Riccati and generalized Riccati transformations, the integral averaging technique, and the theory of comparison. New suﬃcient conditions are established in the noncanonical case. An example is given to support our results

It is notable that in recent years, the oscillation property of solutions of differential equations and their applications have been and still are receiving intensive attention (see [2, 5-7, 15, 18-21]).In the natural sciences, technology, and population dynamics, differential equations find many application fields [1,4,8,9,13].For particular applications of differential equations with neutral term they are often used for the study of distributed networks containing lossless transmission lines [10].Moreover, for particular applications in superlinear wave equation [11].Here, we mention some recent works concerned with special cases of (1.1), which motivated this work.Many authors have studied the oscillatory behavior of solutions of the differential equations where it is notable that some of their results can be extended to the following equations In [24], Zhang et al. studied (1.4) in a noncanoncail case as (1.2).They established new oscillation criteria claiming that it could not be applied in case g(t) = t.More recently, Zhang et al. [22] suggested some new oscillation criteria for the even-order delay differential equation (1.4) with the same noncanonical case for which they stressed that the study of oscillatory properties in this case brings in additional difficulties.Meanwhile, Li and Rogovchenko [14] discussed the oscillatory behavior of a class of even-order neutral differential equations of the form with w(t) = y(t) + p(t)y(g(t)).Their new theorems complement and improve a number of results reported in the literature.In [7], Elabbasy et al. studied the even-order neutral differential equation with several delays where dt = ∞.They used the Riccati substitution technique and comparison with delay equations of the first order to establish new oscillation criteria, which simplify and complement some related results in the literature.
In [9], Grace et al. studied the oscillation of the higher-order dynamic equation with superlinear neutral term where y(t) = x(t) + p(t)x α (g(t)), a (μ) ≥ 0, β ≤ α, and α ≥ 1.Their proposed results provide a unified platform that adequately covers discrete and continuous equations and further sufficiently comments on the oscillation of a more general class of equations than the ones reported in the literature.More recently, Dharuman et al. [3], were concerned with the oscillatory behavior of solutions of the even-order nonlinear differential equation with a superlinear neutral term b(t)z (n-1) (t) + q(t)x β σ (t) = 0, (1.8) where They established new comparison theorems that compare the higher-order equation (1.8) with a couple of first-order delay differential equations.Moreover, as many results are available in the literature on the oscillation of first-order delay differential equations, it would be possible to formulate many criteria for the oscillation of (1.8) based on their results.
In this article, we study the oscillatory behavior of solutions of Eq. (1.1) in the noncanonical case (1.2) by applying the Riccati and generalized Riccati transformations, the integral averaging technique, and comparison theory.

Auxiliary lemmas
In this section, we outline some lemmas needed for our results.
Lemma 3 [22] Let w ∈ C I ([t 0 , ∞), R + ).If w (I) (t) is eventually of one sign for all large t, then there exists t 1 ≥ t 0 and an integer i, 0 ≤ i ≤ I with I + i even for w (I) (t) ≥ 0, or I + i odd for w γ , where B > 0, A, and D are constants.Then, the maximum value of f on R at Lemma 5 Assume that x(t) is an eventually positive solution of (1.1).Then, there exists t 1 ≥ t 0 such that for t ≥ t 1 the corresponding function w satisfies one of the following four cases > 0 for every odd integer i ∈ {1, 2, . . ., n -3} and w (n-1) < 0, r w (n-1) γ ≤ 0.

Main results
Lemma 6 Assume that x(t) is an eventually positive solution of (1.1) with w satisfying case 1 of Lemma 5, then for all constants k 1 > 0, where P ] ≥ 0.
Using the above inequality in the definition of w(t), we get Since ξ (t) ≤ t , ξ (t) is strictly increasing and t ≤ ξ -1 (t), then Since w(t) satisfies case 1 of Lemma 5, so from Lemma 2, we obtain The function t -n+1 w(t) is nonincreasing, which with (3.3) leads to Thus, using (3.2), we obtain .
Hence since w(t) is positive and increasing, there exists a positive constant k 1 such that w(t) ≥ k 1 .Thus, we have Substituting into (1.1),we have and since ξ -1 (t), w (t) are increasing functions, then we have Lemma 7 Assume that x(t) is an eventually positive solution of (1.1) with w satisfying case 2 of Lemma 5, then for all constants k 2 > 0, ) ] ≥ 0.
Lemma 8 Suppose that x(t) is an eventually positive solution of (1.1) with w satisfying case 3 of Lemma 5, then for all constants k 3 > 0, where P ] ≥ 0.
Since w (n-1) (t) is positive and nonincreasing function, there exists a positive constant c 1 such that w (n-1) (t) ≤ c 1 .Then,
By completing the squares .
Integrating from t 2 to t, we get This is a contradiction with (3.6).Assume that we have case 2, using Lemma 1, we find (3.11).Thus, using (3.4), we have we see that z(t) = r(t)(w (n-1) (t)) γ is a positive solution of the differential inequality using [[17], Corollary 1], we see that (3.7) has a positive solution, and this is a contradiction.Assume that we have case 3, then, as in Lemma 8, we have that t 2-n w(t) is nonincreasing, and since h(t) < t, we have [h(t)] 2-n w(h(t)) > t 2-n w(t).Now, we define then φ(t) < 0, and by using (3.5), .
On the other hand, using Lemma 1, we get for every N ∈ (0, 1), and all sufficiently large t, then , since w (n-2) is positive and decreasing then w (n-2) (h(t)) ≥ w (n-2) (t).There exists a positive constant c 2 such that w (n-2) (t) ≤ c 2 , then we have Multiplying (3.13) by E γ (t) and integrating it from t 1 to t, we have

Now using the inequality
we have Noting that r[w (n-1) ] γ is nonincreasing, we have Dividing by r 1 γ (s) and integrating from t to κ gives Letting κ → ∞, we have then By substituting into (3.14),we have ds ≤ E γ (t 1 )φ(t 1 ) + 1, which contradicts (3.8).Assume that we have case 4. Since r[w (n-1) ] γ is nonincreasing, as in the proof of case 3, we get (3.15).Hence, there exists a constant c 3 > 0, such that which contradicts (3.9), and this completes the proof.
From the definition of ϕ(t), we see that This leads to then (1.1) is oscillatory.

Conclusion 13
In this work, we use techniques of the Riccati and generalized Riccati transformations, integral averaging, and the method of comparison to establish some new oscillation criteria for the even-order differential equation with superlinear neutral term (1.1) in noncanonical case.The obtained results improve and complement some previous criteria in the literature.An example is provided to support the theoretical findings.