Asymptotic Behavior of Solutions of Even-Order Advanced Differential Equations

In this paper, we establish the qualitative behavior of the even-order advanced differential equation (a(υ)(y(κ− 1)(υ))β)′ + 􏽐ji�1 qi(υ)g(y(ηi(υ))) � 0, υ≥ υ0. +e results obtained are based on the Riccati transformation and the theory of comparison with firstand second-order equations. +is new theorem complements and improves a number of results reported in the literature. Two examples are presented to demonstrate the main results.


Introduction
Advanced differential equations are of practical importance, which model a phenomenon in which the rate of change of a quantity depends on present and future values of the quantity. Myschkis was the first, who discussed such equations in 1955 [1] and after him Cooke and Bellman worked further on it in 1963 [2]. ese types of equations have been used in modeling of various physical and engineering phenomena. For example, population genetics [3], the study of wavelets [4], population growth [5], the field of time symmetric electrodynamics [6], neural networks [7], optimal control problems with delay [8], economics [8], dynamical systems, mathematics of networks, optimization, electrical power systems, materials, energy j ≥ 1, etc. [9] have been studied using advanced differential equations and many approaches discussed in [10][11][12][13][14][15][16][17][18][19][20][21][22] can be presented for solution of such equations.
In 1980, Shah et al. [23] discussed the uniqueness and existence of the solution to nonlinear and linear such types equations, while the oscillation properties of the solution were investigated by Ladas and Stavroulakis [24], and after that, particularly in the last decade, Further refinements and improvements in the theory of advanced differential equations have been made by different researchers and it is still an active of research in engineering and applied sciences. e present paper deals with the investigation of the qualitative behavior of even-order advanced differential equation: where j ≥ 1 and β are a quotient of odd positive integers. roughout this work, we suppose that By a solution of (1), we mean a function y ∈ C κ− 1 [υ y , ∞), υ y ≥ υ 0 , which has the property a(υ)(y (κ−1) (υ)) β ∈ C 1 [υ y , ∞), and satisfies (1) on [υ y , ∞). We consider only those solutions y of (1) which satisfy sup |y(υ)|: υ ≥ υ y > 0. A solution of (1) is called oscillatory if it has arbitrarily large zeros on [υ y , ∞); otherwise, it is called nonoscillatory. Equation (1) is said to be oscillatory if all of its solutions are oscillatory.
e study of oscillation has been carried to fractional equations in the setting of fractional operators with singular and nonsigular kernels as well (see [46,47] and the references therein).
e main aim of this paper is to complement and improve the results of [48][49]. For this purpose we discuss these results.
Moaaz et al. [26] considered the fourth-order differential equation: where β, α are quotients of odd positive integers. Grace et al. [27] considered the equation where η(υ) ≤ υ, β is a quotient of odd positive integers. In particular, by using the comparison technique, the equation has been studied by Agarwal and Grace [48], and they proved it oscillatory if Agarwal and Grace [48] extended the Riccati transformation to obtain new oscillatory criteria for (5) as condition Authors in [50] studied oscillatory behavior of (5) where β � 1 and if there exists a function τ ∈ C 1 ([υ 0 , ∞), (0, ∞)), also, they proved it oscillatory by using the Riccati transformation if To prove this, we apply the previous results to the equation (1) By applying condition (6) in [48], we get (2) By applying condition (7) in [49], we get (3) By applying condition (8) in [50], we get From the above, we find the results in [49] improve results [50]. Moreover, the results in [48] improve results [49,50].
us, the motivation in studying this paper is complement and improve results [48][49][50].
We shall employ the following lemmas.
Lemma 3 (see [34]). Let β be a ratio of two odd numbers, V > 0 and U are constants. en Lemma 4 (see [29]). Suppose that y is an eventually positive solution of (1). en, there exist two possible cases:

Comparison Theorems with Second/First-Order Equations
Theorem 1. Assume that (2) holds. If the differential equations are oscillatory. en every solution of (1) is oscillatory.
Let case (S 2 ) holds. Define Now, integrating (1) from υ to m and using y ′ (υ) > 0, we find Mathematical Problems in Engineering 3 By virtue of y ′ (υ) > 0 and η i (υ) ≥ υ, we get Letting m ⟶ ∞, we see that and so Integrating again from υ to ∞ for a total of (κ − 4) times, we get From (27) and (32), we obtain If we now set ϑ(υ) � k � 1 in (33), then we obtain From [25], we see equation (18) is nonoscillatory, which is a contradiction. eorem 1 is proved. □ Remark 1. It is well known (see [42]) that if then equation where β � 1 is oscillatory. Based on the above results and eorem 1, we can easily obtain the following Hille and Nehari type oscillation criteria for (1) with β � 1.
In the theorem, we compare the oscillatory behavior of (1) with the first-order differential equations: Theorem 3. Assume that (2) holds. If the differential equations are oscillatory, then every solution of (1) is oscillatory.
Proof. Assume the contrary that y is a positive solution of (1). en, 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 see for every θ ∈ (0, 1) and for all large υ. us, if we set then we see that ψ is a positive solution of the inequality.
From [?, eorem 1], we see that the equation (40) also has a positive solution, which is a contradiction.
In the case where (S 2 ) holds, from Lemma 1, we get From (32) and (45), we get Now, we set us, we find ψ is a positive solution of the inequality It is well known (see [?, eorem 1]) that the equation (41) also has a positive solution, which is a contradiction. e proof is complete.

Remark 2.
We compare our result with the known related criteria for oscillation of this equation as follows (Table 1). erefore, our result improves results [48][49][50].

Conclusion
In this article, we study the oscillatory behavior of a class of nonlinear even-order differential equations and establish sufficient conditions for oscillation of an even-order differential equation by using the theory of comparison with first-and second-order delay equations and Riccati substitution technique.
For researchers interested in this field, and as part of our future research, there is a nice open problem which is finding new results in the following case: For all this, there is some research in progress.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that there are no conflicts of interest associated with this publication.