# Asymptotic Behavior of Solutions of Even-Order Advanced Differential Equations

Authors: Omar Bazighifan; Hijaz Ahmad
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

In this paper, we establish the qualitative behavior of the even-order advanced differential equationaυyκ−1υβ′+∑i=1jqiυgyηiυ=0,υ≥υ0. 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.


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## Body

## 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 j≥1, 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:(1)aυyκ−1υβ′+∑i=1jqiυgyηiυ=0,υ≥υ0,where j≥1 and β are a quotient of odd positive integers. Throughout this work, we suppose that C1:a∈C1υ0,∞,ℝ, aυ>0,a′υ≥0C2:qi,ηi∈Cυ0,∞,ℝ, qiυ≥0,ηiυ≥υ, limυ→∞ηiυ=∞,i=1,2,…,jC3:g∈Cℝ,ℝ such that gx/xβ≥k>0, for x≠0 and under the condition(2)∫υ0∞1a1/βsds=∞.By a solution of (1), we mean a function y∈Cκ−1υy,∞,υy≥υ0, which has the property aυyκ−1υβ∈C1υy,∞, and satisfies (1) on υy,∞. We consider only those solutions y of (1) which satisfy supyυ:υ≥υy>0. A solution of (1) is called oscillatory if it has arbitrarily large zeros on υy,∞; 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:(3)aυy″′υβ′+qυyαηυ=0,where β,α are quotients of odd positive integers.Grace et al. [27] considered the equation(4)aυy″υβ″+qυgyηυ=0,where ηυ≤υ, β is a quotient of odd positive integers.In particular, by using the comparison technique, the equation(5)yκ−1υβ′+qυyβηυ=0,has been studied by Agarwal and Grace [48], and they proved it oscillatory if(6)liminfυ⟶∞∫υηυηs−sκ−2∫s∞qυdυ1/βds>κ−2!e.Agarwal and Grace [48] extended the Riccati transformation to obtain new oscillatory criteria for (5) as condition(7)limsupυ⟶∞υβκ−1∫υ∞qsds>κ−1!β.Authors in [50] studied oscillatory behavior of (5) where β=1 and if there exists a function τ∈C1υ0,∞,0,∞, also, they proved it oscillatory by using the Riccati transformation if(8)∫υ0∞τsqs−κ−2!τ′s223−2κsκ−2τsds=∞.To prove this, we apply the previous results to the equation(9)y4υ+q0υ4y2υ=0,υ≥1.(1)
By applying condition (6) in [48], we get(10)q0>25.5.(2)
By applying condition (7) in [49], we get(11)q0>18.(3)
By applying condition (8) in [50], we get(12)q0>1728.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]).
Ifyiυ>0, i=0,1,…,κ, and yκ+1υ<0, then (13)yυυκ/κ!≥y′υυκ−1/κ−1!.Lemma 2 (see [44]).
Suppose thaty∈Cκυ0,∞,0,∞, yκ is of a fixed sign on υ0,∞, yκ not identically zero and there exists a υ1≥υ0 such that (14)yκ−1υyκυ≤0,for all υ≥υ1. If we have limυ⟶∞yυ≠0, then there exists υθ≥υ1 such that (15)yυ≥θκ−1!υκ−1yκ−1υ,for every θ∈0,1 and υ≥υθ.Lemma 3 (see [34]).
Letβ be a ratio of two odd numbers, V>0 and U are constants. Then (16)Ux−Vxβ+1/β≤βββ+1β+1Uβ+1Vβ,V>0.Lemma 4 (see [29]).
Suppose thaty is an eventually positive solution of (1). Then, there exist two possible cases:S1yυ>0,y′υ>0, y″υ>0,yκ−1υ>0,yκυ<0S2yυ>0, yrυ>0, yr+1υ<0 for all odd integer,r∈1,3,…,κ−3, yκ−1υ>0, yκυ<0Forυ≥υ1, where υ1≥υ0 is sufficiently large.


## 3. Comparison Theorems with Second/First-Order Equations

Theorem 1.
Assume that (2) holds. If the differential equations (17)κ−2!a1/βυθυκ−2βy′υβ′+k∑i=1jqiυyβυ=0,(18)y″υ+yυ1κ−4!∫υ∞ς−υκ−4·1aς∫ς∞∑i=1jqisds1/βdς=0.are oscillatory. Then every solution of (1) is oscillatory.Proof.
Assume the contrary thaty is a positive solution of (1). Then, we can suppose that yυ and yηiυ are positive for all υ≥υ1 sufficiently large. From Lemma 4, we have two possible cases S1 and S2.
Let caseS1 holds. Using Lemma 2, we find (19)y′υ≥θ2υκ−2yκ−1υ,for every θ∈0,1 and for all large υ.
Define(20)φυ≔τυaυyκ−1υβyβυ,we see that φυ>0 for υ≥υ1, where τ∈C1υ0,∞,0,∞ and (21)φ′υ=τ′υaυyκ−1υβyβυ+τυayκ−1β′υyβυ−βτυyβ−1υy′υaυyκ−1υβy2βυ.
Using (19) and (20), we obtain (22)φ′υ≤τ+′υτυφυ+τυaυyκ−1υβ′yβυ−βτυθκ−2!υκ−2aυyκ−1υβ+1yβ+1υ≤τ′υτυφυ+τυaυyκ−1υβ′yβυ−βθυκ−2κ−2!τυaυ1/βφυβ+1/β.
From (1) and (22), we obtain (23)φ′υ≤τ′υτυφυ−kτυ∑i=1jqiυyβηiυyβυ−βθυκ−2κ−2!τυaυ1/βφυβ+1/β.
Note thaty′υ>0 and ηiυ≥υ; thus, we find (24)φ′υ≤τ′υτυφυ−kτυ∑i=1jqiυ−βθυκ−2κ−2!τυaυ1/βφυβ+1/β.
If we setτυ=k=1 in (24), then we find (25)φ′υ+βθυκ−2κ−2!a1/βυφυβ+1/β+∑i=1jqiυ≤0.
From [25], we can see that equation (17) is nonoscillatory, which is a contradiction.
Let caseS2 holds. Define (26)ψυ≔ϑυy′υyυ,we see that ψυ>0 for υ≥υ1, where ϑ∈C1υ0,∞,0,∞. By differentiating ψυ, we find(27)ψ′υ=ϑ′υϑυψυ+ϑυy″υyυ−1ϑυψυ2.
Now, integrating (1) from υ to m and using y′υ>0, we find (28)amyκ−1mβ−aυyκ−1υβ=−∫υm∑i=1jqisgyηisds.
By virtue ofy′υ>0 and ηiυ≥υ, we get(29)amyκ−1mβ−aυyκ−1υβ≤−kyβυ∫υu∑i=1jqisds.
Lettingm⟶∞, we see that (30)aυyκ−1υβ≥kyβυ∫υ∞∑i=1jqisds,and so(31)yκ−1υ≥yυkaυ∫υ∞∑i=1jqisds1/β.
Integrating again fromυ to ∞ for a total of κ−4 times, we get (32)y″υ+yυκ−4!∫υ∞ς−υκ−4·kaς∫ς∞∑i=1jqisds1/βdς≤0.
From (27) and (32), we obtain (33)ψ′υ≤ϑ′υϑυψυ−ϑυκ−4!∫υ∞ς−υκ−4·kaς∫ς∞∑i=1jqisds1/βdς−1ϑυψυ2.
If we now setϑυ=k=1 in (33), then we obtain (34)ψ′υ+ψ2υ+1κ−4!∫υ∞ς−υκ−4·1aς∫ς∞∑i=1jqisds1/βdς≤0.
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 if(35)∫υ0∞1aυdυ=∞,liminfυ⟶∞∫υ0υ1asds∫υ∞qsds>14,then equation(36)aυy′υβ′+qυyβgυ=0,υ≥υ0,where β=1 is oscillatory.
Based on the above results and Theorem1, we can easily obtain the following Hille and Nehari type oscillation criteria for (1) with β=1.Theorem 2.
Letβ=k=1. Assume that (2) holds. If (37)∫υ0∞θυκ−2κ−2!aυdυ=∞,(38)liminfυ⟶∞∫υ0υθsκ−2κ−2!asds∫υ∞∑i=1jqisds>14,also, if(39)liminfυ⟶∞υ∫υ0υ1κ−4!∫v∞ς−υκ−4·1aς∫ς∞∑i=1jqisds1/βdςdv>14,for some constant θ∈0,1. 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 equations(40)x′υ+k∑i=1jqiυθυκ−2κ−2!a1/βυβxηυ=0,(41)z′υ+zυυκ−4!∫υ∞ς−υκ−4·kaς∫ς∞∑i=1jqisds1/βdς=0,are oscillatory, then every solution of (1) is oscillatory.Proof.
Assume the contrary thaty is a positive solution of (1). Then, we can suppose that yυ and yηiυ are positive for all υ≥υ1 sufficiently large. From Lemma 4, we have two possible cases S1 and S2.
In the case whereS1 holds, from Lemma 2, we see (42)yυ≥θυκ−2κ−2!a1/βυa1/βυyκ−1υ,for every θ∈0,1 and for all large υ. Thus, if we set (43)xυ=aυyκ−1υβ>0,then we see that ψ is a positive solution of the inequality.(44)x′υ+k∑i=1jqiυθυκ−2κ−2!a1/βυβxηυ≤0.
From [?, Theorem1], we see that the equation (40) also has a positive solution, which is a contradiction.
In the case whereS2 holds, from Lemma 1, we get (45)yυ≥υy′υ.
From (32) and (45), we get (46)y″υ+y′υυκ−4!∫υ∞ς−υκ−4·kaς∫ς∞∑i=1jqisds1/βdς≤0.
Now, we set(47)zυ=y′υ.
Thus, we findψ is a positive solution of the inequality (48)z′υ+zυυκ−4!∫υ∞ς−υκ−4·kaς∫ς∞∑i=1jqisds1/βdς≤0.
It is well known (see [?, Theorem1]) that the equation (41) also has a positive solution, which is a contradiction. The proof is complete.Corollary 1.
Let (2) holds. If (49)liminfυ⟶∞∫ηiυυ∑i=1jqisθυκ−2κ−2!a1/βυβds>κ−1!βeliminfυ⟶∞∫ηiυυsκ−4!∫υ∞ς−υκ−4kaς∫ς∞∑i=1jqisds1/βdςds>1e,then every solution of (1) is oscillatory.Example 1.
Let the equation(50)y4υ+q0υ4y3υ=0,υ≥1,where q0>0 is a constant. Note that β=1,κ=4,aυ=1,qυ=q0/υ4, andηυ=3υ. If we set k=1, then condition (38) becomes (51)liminfυ⟶∞∫υ0υθsκ−2κ−2!asds∫υ∞∑i=1jqisds=liminfυ⟶∞υ33∫υ∞q0s4ds=q09>14,and condition (39) becomes(52)liminfυ⟶∞υ∫υ0υ1κ−4!∫v∞ς−υκ−4·1aς∫ς∞∑i=1jqisds1/βdςdv=liminfυ⟶∞υq06υ=q06>14.
Therefore, from Theorem2, all solution equation (51) is oscillatory if q0>2.25.Remark 2.
We compare our result with the known related criteria for oscillation of this equation as follows (Table1).
Therefore, our result improves results [48–50].Table 1
Comparision of results.
The condition(6)(7)(8)The criterionq0>13.6q0>18q0>576Example 2.
Consider a differential equation (9) where q0>0 is a constant. Note that β=1,κ=4,aυ=1,qυ=q0/υ4, and ηυ=2υ. If we set k=1, then condition (38) becomes (53)q09>14.
Therefore, from Theorem2, all solution equation (9) is oscillatory if q0>2.25.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:(54)∫υ0∞1a1/βsds<∞.For all this, there is some research in progress.



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