Abstract
In this paper, we establish the qualitative behavior of the even-order advanced differential equation . The results obtained are based on the Riccati transformation and the theory of comparison with first- and second-order equations. This new theorem complements and improves a number of results reported in the literature. Two examples are presented to demonstrate the main results.
1. 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]. These 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 , etc. [9] have been studied using advanced differential equations and many approaches discussed in [10–22] 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. The present paper deals with the investigation of the qualitative behavior of even-order advanced differential equation:where and are a quotient of odd positive integers. Throughout this work, we suppose that C1: , C2: , , C3: such that for and under the condition
By a solution of (1), we mean a function , which has the property , and satisfies (1) on . We consider only those solutions of (1) which satisfy . A solution of (1) is called oscillatory if it has arbitrarily large zeros on ; otherwise, it is called nonoscillatory. Equation (1) is said to be oscillatory if all of its solutions are oscillatory.
2. The Motivation of Studying this Paper
During this decade, several works have been accomplished in the development of the oscillation theory of higher-order advanced equations by using the Riccati transformation and the theory of comparison between first- and second-order delay equations [25–39]. Further, the oscillation theory of fourth- and second-order equations has been studied and developed by using integral averaging technique and the Riccati transformation [40–45].
The 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).
The 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 are quotients of odd positive integers.
Grace et al. [27] considered the equationwhere , is a quotient of odd positive integers.
In particular, by using the comparison technique, the equationhas 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 and if there exists a function , 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].
Thus, the motivation in studying this paper is complement and improve results [48–50].
We shall employ the following lemmas.
Lemma 1 (see [44]). If , , and , then
Lemma 2 (see [44]). Suppose that , is of a fixed sign on , not identically zero and there exists a such that for all . If we have , then there exists such that for every and .
Lemma 3 (see [34]). Let be a ratio of two odd numbers, and are constants. Then
Lemma 4 (see [29]). Suppose that is an eventually positive solution of (1). Then, there exist two possible cases: , , , for all odd integer,, ,
For where is sufficiently large.
3. Comparison Theorems with Second/First-Order Equations
Theorem 1. Assume that (2) holds. If the differential equations are oscillatory. Then every solution of (1) is oscillatory.
Proof. Assume the contrary that is a positive solution of (1). Then, we can suppose that and are positive for all sufficiently large. From Lemma 4, we have two possible cases and .
Let case holds. Using Lemma 2, we find for every and for all large .
Definewe see that for , where and Using (19) and (20), we obtain From (1) and (22), we obtain Note that and ; thus, we find If we set in (24), then we find From [25], we can see that equation (17) is nonoscillatory, which is a contradiction.
Let case holds. Define we see that for where . By differentiating , we findNow, integrating (1) from to and using , we find By virtue of and , we getLetting , we see that and soIntegrating again from to for a total of times, we get From (27) and (32), we obtain If we now set in (33), then we obtain From [25], we see equation (18) is nonoscillatory, which is a contradiction. Theorem 1 is proved.
Remark 1. It is well known (see [42]) that ifthen equationwhere is oscillatory.
Based on the above results and Theorem 1, we can easily obtain the following Hille and Nehari type oscillation criteria for (1) with .
Theorem 2. Let . Assume that (2) holds. If also, iffor some constant . Then all solution of (1) is oscillatory.
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 equationsare oscillatory, then every solution of (1) is oscillatory.
Proof. Assume the contrary that is a positive solution of (1). Then, we can suppose that and are positive for all sufficiently large. From Lemma 4, we have two possible cases and .
In the case where holds, from Lemma 2, we see for every and for all large . Thus, if we set then we see that is a positive solution of the inequality.From [?, Theorem 1], we see that the equation (40) also has a positive solution, which is a contradiction.
In the case where holds, from Lemma 1, we get From (32) and (45), we get Now, we set Thus, we find is a positive solution of the inequality It is well known (see [?, Theorem 1]) that the equation (41) also has a positive solution, which is a contradiction. The proof is complete.
Corollary 1. Let (2) holds. If then every solution of (1) is oscillatory.
Example 1. Let the equation where is a constant. Note that , and. If we set , then condition (38) becomes and condition (39) becomesTherefore, from Theorem 2, all solution equation (51) is oscillatory if .
Remark 2. We compare our result with the known related criteria for oscillation of this equation as follows (Table 1).
Therefore, our result improves results [48–50].
Example 2. Consider a differential equation (9) where is a constant. Note that , and . If we set , then condition (38) becomes Therefore, from Theorem 2, all solution equation (9) is oscillatory if .
Remark 3. Our result improves results [48–50].
4. 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
The authors declare that there are no conflicts of interest associated with this publication.