Multiple Techniques for Studying Asymptotic Properties of a Class of Differential Equations with Variable Coefﬁcients

: This manuscript is concerned with the oscillatory properties of 4th-order differential equations with variable coefﬁcients. The main aim of this paper is the combination of the following three techniques used: the comparison method, Riccati technique and integral averaging technique. Two examples are given for applying the criteria.

In the last few decades, there have been a constant interest to investigate the asymptotic property for oscillations of differential equation, see . Furthermore, there are some results that study the oscillatory behavior of 4th-order equations with p-Laplacian, we refer the reader to [26,27]. Now the following results are presented. Grace and Lalli [28], Karpuz et al. [29] and Zafer [30] studied the even-order equation they used the Riccati substitution to find several oscillation criteria and established the following results, respectively: where and lim inf Zhang et al. [31,32] studied the even-order equation where β is a quotient of odd positive integers. They proved that it is oscillatory, if where γ ≥ 2 is even and they used the compare with first order equations. If there exists a function for some constant θ ∈ (0, 1) . Our aim in this work is to complement results in [28][29][30][31][32]. Two examples are given for applying the criteria.

Definition 2. Let
A kernel function H i ∈ C (D, R) is said to belong to the function class , written by H ∈ , if, for i = 1, 2, has a continuous and nonpositive partial derivative ∂H i /∂s on D 0 and there exist functions and and are oscillatory, then every solution of (1) is oscillatory.
Integrating again from ζ to ∞, we get Combining (20) and (21), we find If ϑ (ζ) = m = 1 in (22), we get Thus, the Equation (13) is nonoscillatory, which is a contradiction. The proof of the theorem is complete.
Next, we obtain the following Hille and Nehari type oscillation criteria for (1) with p = 2.
In this theorem, we use the integral averaging technique: and lim sup where for all θ ∈ (0, 1) , and then (1) is oscillatory.
Proof. Proceeding as in the proof of Theorem 1. Assume that (S 1 ) holds. From Theorem 1, we get that (18) holds. Multiplying (18) by H 1 (ζ, s) and integrating the resulting inequality from ζ 1 to ζ, we find that From (10), we get and so 1 which contradicts (26). The proof of the theorem is complete.

Remark 2.
We point out that continuing this line of work, we can have oscillatory results for a fourth order equation of the type: under the condition ∞ ζ 0 1 a 1/(p−1) (s) ds < ∞.

Conclusions
In this article, we studied some oscillation conditions for 4th-order differential equations by the comparison method, Riccati technique and integral averaging technique.
Further, in the future work we study Equation (1)