On the Oscillation Criteria for Fourth-Order p-Laplacian Differential Equations with Middle Term

In this paper, we study the oscillatory properties of the solutions of a class of fourth-order p-Laplacian differential equations with middle term. The new oscillation criteria obtained by using the theory of comparison with firstand second-order differential equations and a refinement of the Riccati transformations. The results in this paper improve and generalize the corresponding results in the literatures. Three examples are provided to illustrate our results.

In this paper, motivated by [26][27][28], we will give some new sufficient conditions for that oscillatory behavior of (1). In Section 2, we will provide some lemmas that will help us to prove our main results. In Section 3, based on the comparison with firstand second-order differential equations and a refinement of the Riccati transformations, we establish some new oscillation criteria of (1).

Preliminaries
First, we give the following lemmas that can discuss our main results.

Lemma 4 ([31], Lemma 2.3).
Assume that α is a quotient of odd positive integers; V > 0 and U ∈ ℝ are constants. Then The following lemma will be used in the proof of our main results in the next section.

Main Results
In the following theorem, we then by using a comparison strategy involving first-order differential equations to provide an oscillation criterion for equation (1).

For convenience, let
Theorem 7. Assume that (H 1 ) and hold. If the differential equation is oscillatory for some μ ∈ ð0, 1Þ, then the differential equation (1) is oscillatory.
Proof. Assume that (1) has a nonoscillatory solution in ½t 0 , ∞Þ. Without loss of generality, we may let x be an eventually positive solution of (1). Then, there exists a t 1 ≥ t 0 such that xðtÞ > 0, xðτðtÞÞ > 0, and xðσðtÞÞ > 0 for t ≥ t 1 . Let which having in mind (1) gives From the definition of zðtÞ, one has By repeating the same process, we have Set n = 3 in Lemma 2, we obtain zðtÞ ≥ ð1/3Þtz ′ ðtÞ, which implies that zðtÞ/t 3 is nonincreasing. Moreover, by the fact τðtÞ ≤ t that gives Combining (24) and (25), which yields Between equations (1) and (26), we obtain Since z is positive and increasing (by Lemma 5), we have lim t⟶∞ zðtÞ=0. So, from Lemma 3, one has for some μ ∈ ð0, 1Þ. It follows between (27) and (28) that, for all μ ∈ ð0, 1Þ, ω is a positive solution of the first-order delay differential inequality

Journal of Function Spaces
It is well known (see [33] and Theorem 7) that the corresponding equation (20) also has a positive solution, which is a contradiction. The theorem is proved.

Journal of Function Spaces
It is well known (see [33] and Theorem 7) that the corresponding equation (31) also has a positive solution, which is a contradiction. The theorem is proved.
Proof. Our proof by reduction to the absurd. Assume that z ′ ′ðtÞ > 0. From Lemma 2, we obtain Integrating the above equality from σðtÞ to t, one find that Let hðtÞ = z′ðtÞ in Lemma 3, then for all ε 1 ∈ ð0, 1Þ and every sufficiently large t. Now, we define a function ϕ by By differentiating (41) and using the inequalities (39) and (40), we get Since z′ðtÞ > 0, there exist a t 2 ≥ t 1 and a constant M > 0 such that zðtÞ > M, for all t ≥ t 2 . Without loss of generality, we may let M ≥ 1. By using Lemma 4 with we obtain

Journal of Function Spaces
This implies that which contradicts (37). The proof is completed.
Proof. We use the reduction to the absurd arguments. Assume that (1) has a nonoscillatory solution in ½t 0 , ∞Þ. Without loss of generality, we only need to be concerned with positive solutions of equation (1). Then, there exists a t 1 ≥ t 0 such that xðtÞ > 0, xðτðtÞÞ > 0, and xðσðtÞÞ > 0 for t ≥ t 1 . From Lemmas 4 and 11, one has that