Oscillation of super-linear fourth-order differential equations with several sub-linear neutral terms

In this paper, we discuss the oscillatory behavior of solutions of a class of Super-linear fourth-order differential equations with several sub-linear neutral terms using the Riccati and generalized Riccati transformations. Some Kamenev–Philos-type oscillation criteria are established. New oscillation criteria are deduced in both canonical and non-canonical cases. An illustrative example is given.

Oscillation phenomena take part in different models from real-world applications; see, e.g., paper [8] for more details. In the last three decades, there has been considerable interest in studying the oscillation of solutions of several kinds of differential equations [1-5, 7, 8, 10-20, 22-24, 26-39]. The half-linear equations have numerous applications in the study of p-Laplace equations, non-Newtonian fluid theory, porous medium, etc.; see, e.g., papers [6,21,25] for more details. In particular, papers [11,24] were concerned with the oscillation of various classes of half-linear differential equations, whereas the papers [3-5, 7, 10, 20, 26, 38] were concerned with the oscillatory behavior of the fourth-order differential equation (1.1) and its special cases. In what follows, we briefly comment on a number of closely related results which motivated our work. The authors in [3,4,26] discussed in their recent papers, the special case of (1.1) of the form, Under the condition (1.2), Dassios and Bazighifan in [10] discussed the oscillation of the same equation under condition (1.3). In [20], Li et al. studied the oscillatory behavior of a class of fourth-order differential equations with the p-Laplacian-like operator of the type, dt < ∞, they used the Riccati transformation and integral averaging technique and presented a Kamenev-type oscillation criterion. More recently, Bazighifan et al. [5] studied the asymptotic behavior of solutions of the fourth-order neutral differential equation with the continuously distributed delay of the form where α, β are quotients of odd positive integers, and β ≥ α under the condition (1.2).

Preliminaries
The following preliminary results will be needed for our proofs.

Lemma 4 ([2])
Let α is a ratio of two odd numbers. Suppose that U, V are constants with

Lemma 5
Assume that x(t) is an eventually positive solution of (1.1), z (t) > 0, and there exists a positive decreasing function Proof Let x be an eventually positive solution of Eq. (1.1). Then, there exists a t 1 ≥ t 0 such that x(t) > 0, x(σ j (t)) > 0 and x(τ i (t)) > 0 for t ≥ t 1 . Now from the definition of z, we have Then, by Lemma 1, we have Now since z(t) is positive and increasing, and δ(t) is a positive decreasing function tending to zero, then there exists a t 2 ≥ t 1 such that z(t) ≥ δ(t), and That is x(t) ≥ θ (t)z(t). Therefore, from (1.1), it follows that Thus, the proof is completed.
The following two auxiliary results are very similar to those reported in [3] and [10].

Main results
We first consider the case R(t 0 ) = ∞.
Then, by Lemma 4, we obtain Thus, we have which contradicts (3.1), and this completes the proof.
The following result deals with the Kamenev-type oscillation for Eq. (1.1) under the condition (1.2).
Proof Let x be a nonoscillatory solution of (1.1) on [t 0 , ∞). Without loss of generality, we may assume that x is an eventually positive solution. Define ψ(t) as in (3.2). Then, following the same steps as in the proof of Theorem 8, we arrive at (3.6). Multiplying (3.6) by (ts) n and integrating the resulting inequality from t 0 to t, we have [ζ τ (s)τ 2 (s)] γ ds. Hence, and so lim sup which contradicts (3.7), and this completes the proof. Now we are going to discuss the so called Philos-type oscillation criteria for Eq. (1.1) under condition (1.2), but we first outline the following definition. (D, R), i = 1, 2 are said to belong to the class X (written K i ∈ X) if they satisfy and

Theorem 11
Assume that there exists a function K 1 ∈ X such that lim sup Then, Eq. (1.1) is oscillatory.
Proof Let x be a nonoscillatory solution of (1.1). Without loss of generality, we may assume that x is an eventually positive solution of (1.1). Now define ψ(t) as in (3.2). Following the same steps as in the proof of Theorem 8, we arrive at (3.5). Multiplying (3.5) by K 1 (t, s) and integrating the resulting inequality from T to t, we have where .
Then, we have Putting U = |L 1 (t, s)|, V = K 1 (t, s)B(s) and then using Lemma 4, we obtain for all sufficiently large t, which contradicts (3.9).
Proof Assume that x(t) is an eventually positive solution of (1.1). Then, there exists a t 1 ≥ t 0 such that x(t) > 0, x(σ j (t)) > 0 and x(τ i (t)) > 0 for t ≥ t 1 . Using Lemma 5, we arrive at (2.1). Define Then, it is clear by (2.1) that Since, by Lemma 2, we have then i.e.
Integrating the above inequality from t to l, we get Letting l → ∞ and using the fact that ω(t) is positive and decreasing, we get Then obviously δ ≥ 1, and by (3.10) and (3.11), it follows that which contradicts the admissible values of δ ≥ 1 and γ ≥ 1. Therefore, the proof is completed.

The case R(t 0 ) < ∞
Now we are going to discuss the oscillatory behavior of Eq. (1.1) under the condition (1.3). First we need the following lemma.
The proof is completed. Proof Suppose for the contrary that there exists a nonoscillatory solution x(t) > 0 of (1.1). Then, we have one of the three possible cases of Lemma 7. We first assume that (S 1 ) holds. Then by Theorem 11, if (3.9) holds, Eq. (1.1) is oscillatory. Secondly, if (S 2 ) holds, then by Lemma 13, we get (4.2). Multiplying (4.2) by K 2 (t, s) and integrating from t 1 to t, we obtain t t 1 K 2 (t, s)ρ(s)