An Improved Criterion for the Oscillation of Fourth-Order Differential Equations

: The main purpose of this manuscript is to show asymptotic properties of a class of differential equations with variable coefficients (cid:16) r ( ν ) ( w (cid:48)(cid:48)(cid:48) ( ν )) β (cid:17) (cid:48) + ∑ ji = 1 q i ( ν ) y κ ( g i ( ν )) = 0, where ν ≥ ν 0 and w ( ν ) : = y ( ν ) + p ( ν ) y ( σ ( ν )) . By using integral averaging technique, we get conditions to ensure oscillation of solutions of this equation. The obtained results improve and generalize the earlier ones; finally an example is given to illustrate the criteria.


Introduction
In this paper, we study the oscillatory properties of solutions of the following fourth-order neutral differential equation where j ≥ 1 and w (ν) := y (ν) + p (ν) y (σ (ν)) .
A kernel function H i ∈ C (D, R) is said to belong to the function class if, for i = 1, 2 (i) H i (ν, s) = 0 for ν ≥ ν 0 , H i (ν, s) > 0, (ν, s) ∈ D 0 ; (ii) ∂H i /∂s exists on D 0 and it is continuous and non-positive. Moreover, there exist three functions ξ, π ∈ C 1 ([ν 0 , ∞) , (0, ∞)) and h i ∈ C (D 0 , R) such that and The historical background of neutral differential equations is extremely varied. In fact, they find numerous applications in natural science but also in technology: in the study of distributed networks containing lossless transmission lines, in high-speed computers, in the theory of automatic control and in aeromechanical systems (see [1]). In last years, the asymptotic properties of solutions of differential equations has been the subject of intensive study (see ). The model of human balancing is considered in [21][22][23] where results on stability are presented.
Furthermore, many researchers investigate regularity and existence properties of solutions to partial differential equations. See for instance [11,[24][25][26][27] and the references therein. Also, we mention the study of the exact solutions to partial differential equations performed with a Lie symmetry analysis. A recent result in this direction is represented by [9] where the authors study a modified Schrödinger equation.
Interesting applications of neutral differential equations can be found in the study of the effects of vibrating systems fixed to an elastic bar, for example the Euler equations of the fluid dynamics (see the recent paper [10]).
In [28], the author obtained the necessary and sufficient conditions under which a general fourth-order ordinary differential equation admits a unique Lagrangian. Nevertheless, there exist examples of fourth-order ordinary differential equations which do not have a second-order Lagrangian.
Many papers have been concerned to the solution of the inverse problem of calculus of variations, namely finding a Lagrangian of differential equations. Also, the use of the Jacobi last multiplier and its connection with Lie theory, in order to find the Lagrangian for ordinary differential equations, can be found in [29]. Now we state some preliminary and interesting results related to the contents of this paper. Zafer [33], Zhang and Yan [35] studied the equation where n is even and established some new sufficient conditions for oscillation.
, then every solution of (5) is oscillatory.
Now, we consider the equation By applying condition (6) Thus, we get that (8)

is oscillatory if
The condition (6) (7) The criterion q 0 > 1839.2 q 0 > 59.5 From above, we see that [35] enriched the results in [33]. Thus, the motivation in studying this paper is to extend the already interesting and pioneer results in [33,35]. By using integral averaging technique, new oscillatory criteria for (1) are established. Furthermore, in order to illustrate the criteria presented here, an example is given.

Oscillation Criteria
For convenience, we denote
Proof. Let y be a non-oscillatory solution of (1) on [ν 0 , ∞). Without loss of generality, we can assume that y is eventually positive. It follows from Lemma 4 that there exist two possible cases (N 1 ) and (N 2 ). Assume that (N 1 ) holds. From Lemma 6, we get that (12) holds. Multiplying (12) by H (ν, s) and integrating the resulting inequality from ν 1 to ν, we find that Using (ν, s) and y = ψ (s), we get which with (26) gives which contradicts (24).

Conclusions
This paper deals with a class of fourth-order neutral differential equations with variable coefficients. Using the famous Riccati's transformation, we establish a new asymptotic criterion that improves and complements the findings contained in [33,35]. Moreover, we get Philos type oscillation criteria to ensure oscillation of solutions of the Equation (1). Furthermore, in a future work we will get some oscillation criteria for (1) under the condition Funding: This research received no external funding.