Improved Approach for Studying Oscillatory Properties of Fourth-Order Advanced Differential Equations with p-Laplacian Like Operator

This paper aims to study the oscillatory properties of fourth-order advanced differential equations with p-Laplacian like operator. By using the technique of Riccati transformation and the theory of comparison with first-order delay equations, we will establish some new oscillation criteria for this equation. Some examples are considered to illustrate the main results.


Introduction
In the last decades, many researchers from all fields of science, technology and engineering have devoted their attention to introducing more sophisticated analytical and numerical techniques to solve and analyze mathematical models arising in their fields.
Fourth-order advanced differential equations naturally appear in models concerning physical, biological, chemical phenomena applications in dynamical systems, mathematics of networks,and optimization. They also appear in the mathematical modeling of engineering problems to study electrical power systems, materials and energy, elasticity, deformation of structures, and soil settlement [1]. The p-Laplace equations have some applications in continuum mechanics, see for example [2][3][4].
An active and essential research area in the above investigations is to study the sufficient criterion for oscillation of delay differential equations. In fact, during this decade, Several works have been accomplished in the development of the oscillation theory of higher order delay and advanced equations by using the Riccati transformation and the theory of comparison between first and second-order delay equations, (see [5][6][7][8][9][10][11][12]). Further, the oscillation theory of fourth and second order delay equations has been studied and developed by using integral averaging technique and the Riccati transformation, (see [13][14][15][16][17][18][19][20][21][22][23][24][25][26][27]). The study of oscillation has been carried to fractional equations in the setting of fractional operators with singular and nonsingular kernels, as well (see [28,29] and the references therein).
We provide oscillation properties of the fourth order advanced differential equation with a p-Laplacian like operator where υ ≥ υ 0 and j ≥ 1. Throughout this paper, we assume that: In fact, our aim in this paper is complete and improves the results in [5][6][7]. For the sake of completeness, we first recall and discuss these results. Li et al. [3] examined the oscillation of equation where p > 1 is a real number. The authors used the Riccati transformation and integral averaging technique. Park et al. [8] used Riccati technique to obtain necessary and sufficient conditions for oscillation of where κ is even and under the condition Agarwal and Grace [5] considered the equation where κ is even and they proved it oscillatory if Agarwal et al. in [6] studied Equation (4) and obtained the criterion of oscillation Authors in [7] studied oscillatory behavior of (4) where γ = 1, κ is even and if there exists a function δ ∈ C 1 ([υ 0 , ∞) , (0, ∞)) , also, they proved it oscillatory by using the Riccati transformation if To compare the conditions, we apply the previous results to the equation 1. By applying condition (5) on Equation (8), we get q 0 > 13.6.
From the above we find the results in [6] improves results [7]. Moreover, the results in [5] improves results [6,7], we see this clearly in the Section 3. Thus, the motivation in studying this paper is complement and improve results [5][6][7].
We will need the following lemmas.

Lemma 3 ([21]
). Let γ be a ratio of two odd numbers, V > 0 and U are constants. Then

Lemma 4 ([15]).
Assume that y is an eventually positive solution of (1). Then, there exist two possible cases:

Oscillation Criteria
In this section, we shall establish some oscillation criteria for Equation (1).

Lemma 5.
Assume that y be an eventually positive solution of (1) and (S 1 ) holds. If where for all υ > υ 1 , where υ 1 large enough.
Proof. Let y is an eventually positive solution of (1) and (S 2 ) holds. Integrating (1) from υ to m and using y (υ) > 0, we obtain By virtue of y (υ) > 0 and Letting m → ∞ , we see that Integrating again from υ to ∞, we get From the definition of ξ (υ), we see that ξ (υ) > 0 for υ ≥ υ 1 . By differentiating, we find From (15) and (16), we obtain The proof is complete.
In the next theorem, we compare the oscillatory behavior of (1) with the first-order differential equations: Theorem 2. Assume that (3) holds. If the differential equations and are oscillatory, then every solution of (1) is oscillatory.
Proof. Assume the contrary that y is a positive solution of (1). Then, we can suppose that y (υ)and y (η i (υ)) are positive for all υ ≥ υ 1 sufficiently large. From Lemma 4, we have two possible cases (S 1 ) and (S 2 ).
In the case where (S 1 ) holds, from Lemma 2, we get for every ε ∈ (0, 1) and for all large υ. Thus, if we set then we see that ξ is a positive solution of the inequality From ( [27], Theorem 1), we conclude that the corresponding Equation (25) also has a positive solution, which is a contradiction.
In the case where (S 2 ) holds, from Lemma 1, we get From (28) and (15), we get Thus, if we set then we see that ξ is a positive solution of the inequality It is well known (see ([27], Theorem 1)) that the corresponding Equation (26) also has a positive solution, which is a contradiction. The proof is complete. and then every solution of (1) is oscillatory.

Remark 1.
We compare our result with the known related criteria for oscillation of this equation are as follows: The condition (5) (6) (7) our condition The criterion q 0 > 25.5 q 0 > 18 q 0 > 1728 q 0 > 4.5 Therefore, it is clear that we see our result improves results [5][6][7].
For an application of Theorem 1, we give the following example.