# Asymptotic dichotomy in a class of higher order nonlinear delay differential equations

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## Abstract

## Keywords

Asymptotic behavior delay differential equation higher order differential equation oscillation Schwarz inequality## MSC

34K11 34K25## 1 Introduction

*n*is a positive integer, \(I= [ {a,+\infty } ) \subset \mathbb{R}\) (\(a\ge 0\)). The coefficients \(p\in C^{1} ( {I,\mathbb{R}} )\) and \(q\in C ( {I,\mathbb{R}} )\) satisfy \(p(t)\ge 0\) and \(q(t)>0\). The function \(g\in C^{1}({I, \mathbb{R}})\) satisfies \(0< g ( t )\le t\), \({g}' ( t ) \ge 0\), and \(\mathop{\lim }_{t\to +\infty } g ( t )= + \infty \). The function \(f\in C ( {\mathbb{R},\mathbb{R}} )\) satisfies \(f ( u )/u\ge K\) (\(u\ne 0\)) for some positive constant

*K*. Our attention is restricted to the solutions of equation (1.1) which exist on the interval

*I*and satisfy \(\sup_{t\ge T}|y(t)|>0\) for any \(T \ge a\). We make a standing hypothesis that equation (1.1) possesses such solutions.

Since Sturm [20] introduced the concept of oscillation when he studied the problem of the heat transmission, the oscillation theory has been a very active area of research in the qualitative analysis of both ordinary and functional differential equations. Usually, a qualitative approach is concerned with the behavior of solutions of a given differential equation and does not seek explicit solutions. Since then, asymptotic and oscillatory properties of solutions to different equations, functional differential equations, and dynamical equation have attracted the attention of many researchers.

The oscillation and asymptotic behavior have extensive applications in the real world, the readers can refer to the monographs [1, 4, 6, 14], and [21] for more details. The problem of obtaining the oscillatory and asymptotic behavior of certain higher order nonlinear functional differential equations has been studied by a number of authors. The interested readers can see [3, 5, 9, 10, 11, 13, 17, 18], and the references cited therein.

There are many excellent works studying the oscillations and asymptotic behaviors of solutions to higher-order nonlinear delay differential equations, to list all of which is almost impossible. We just list some studies relating to our work below.

*n*-order nonlinear functional differential equation:

*α*is the ratio of two positive odd integers. The oscillation theorems established in that paper extend a number of existing results.

The present paper is organized as follows. In Sect. 2, we present some lemmas which are useful in the proof of our main results. In Sect. 3, we carry out delicate analysis to give several oscillatory and asymptotic criteria for the higher order nonlinear delay differential equation (1.1). Noting that the delay \(g(t)\) has the form \(g(t)=t-\tau \) or the form \(g(t)=at\) in many applications, therefore, in Sect. 4, we give two examples to illustrate the applications of our main theorems.

## 2 Some preliminary lemmas

To give the main results of this paper, we first present and prove some useful lemmas. These lemmas play central roles in the proof of our new oscillation and asymptotic results in the next section.

### Lemma 2.1

*Suppose that the equation*

*is non*-

*oscillatory and suppose that*\(y ( t )\)

*is a non*-

*oscillatory solution of equation*(1.1)

*on*\([T,+\infty )\),

*where*\(T\ge a\).

*Then there exists*\(t_{0} \in [T,+\infty )\)

*such that*,

*for any*\(t\ge t_{0} \),

*we have*\(y^{ ( n )}(t)>0\)

*or*\(y^{ ( n )}(t)<0\).

### Proof

*a*and

*b*(\(t_{0} < a< b\)) are its two adjacent zero points of

*x*such that \({x}'(a)\ge 0\) and \({x}'(b)\le 0\). From equation (2.1) and equation (2.2), we get

*a*to

*b*on both sides of equation (2.3), we obtain

### Lemma 2.2

*Suppose that equation* (2.1) *is non*-*oscillatory*, \(y ( t )\)*is a non*-*oscillatory solution to equation* (1.1), *and there exists a constant*\(T\ge a\)*such that*\(y(t)y^{ ( n )}(t)>0\)*for*\(t\ge T\ge a\). *Then*\(y(t)y^{ ( {n+1} )} ( t )\)*is eventually positive*.

### Proof

From the above inequality, we know \(y^{ ( {n+1} )} ( t )\) is strictly monotonically decreasing in the interval \([T,+\infty )\), and therefore \(y^{ ( {n+1} )} ( t )\) is eventually positive or eventually negative.

*T*to

*t*on both sides of inequality (2.5), we obtain

By careful check of the proving process of Lemma 1.1 in [8], we obtain the following results which will be used in the following proof. □

### Lemma 2.3

*Assume that*\(x\in C^{n} ( {[a,+\infty ),\mathbb{R}} )\)

*such that*\(x ( t )>0\), \(x^{ ( n )} ( t ) \le 0\)

*for*\(t\ge a\),

*and*\(x^{ ( n )} ( t )\)

*does not vanish identically on any interval*\([ {T,\infty } ) \subset [a,+\infty )\).

*If*

*n*

*is even*(

*or odd*),

*then there exists*\(l\in \{ {1, 3,\ldots , n-1} \}\) (

*respectively*\(l\in \{ {0,2,4,\ldots , n-1} \}\))

*such that*,

*for all sufficiently large*

*t*, \(x ( t ) x^{ ( j )} ( t )>0\)

*for*\(j=0, 1,\ldots , l\)

*and*\(( {-1} )^{n+j-1}x ( t ) x^{ ( j )} ( t )>0\)

*for*\(j=l+1, l+2,\ldots , n-1\).

*Furthermore*,

*if*\(l\ge 1\),

*then*

*for all sufficiently large*

*t*.

### Lemma 2.4

*Suppose that the function*\(x\in C^{2} ( {[T,+\infty ), \mathbb{R}} )\),

*then both of the following statements hold*:

- (i)
*If*\(x''(t) \le 0\)*and*\({x}' ( t )<0\)*for*\(t\ge T\),*then*\(\lim_{t\to +\infty } x(t)=-\infty \); - (ii)
*If*\(\lim_{t\to +\infty } x'(t)=\mu \),*where*\(\mu >0\)*or*\(\mu =+\infty \),*then*\(\lim_{t\to +\infty } x(t) = +\infty \).

The statements are obvious, which can be easily checked, we therefore omit the proof here.

### Lemma 2.5

*Suppose that*

*n*

*is a positive integer*, \(x\in C^{n}( {[T,+\infty ), \mathbb{R}})\), \(g\in C^{1}({[T,+\infty ), \mathbb{R}})\), \(x^{(n )}(t )<0\), \(x(t )>0\), \(g(t )< t\)

*for*\(t\ge T\)

*and*\(\lim_{t\to +\infty } x(t) \ne 0\), \(\lim_{t\to +\infty } g(t )=+\infty \).

*Then both of the following statements hold*:

- (i)
*If*\(n=1\),*then there exist*\(\lambda >0\)*and*\(T'>0\)*such that*\(x({g(t)})>x(t )>\lambda \)*for*\(t\ge T'\); - (ii)
*Suppose that*\(n\ge 2\),*then there exist*\(\lambda >0\)*and*\({T}'>0\)*such that*\(0< x^{({n-1})}(t )<\lambda <x(t )\), \(x({g(t )})> \lambda \)*for*\(t\ge {T}'\).

### Proof

Suppose first \(n=1\). Then \(x'(t)<0\) and \(x(t)>0\) for \(t\ge T\). Hence the function \(x(t )>0\) is monotonically decreasing on the interval \([T,+\infty )\). Using the monotone bounded theorem, we know the limit \(\lim_{t\to +\infty } x(t)\) exists and we denote \(\lim_{t\to +\infty } x(t)=\lambda \). Using the monotonicity of \(x(t)\) again, we know \(x(t)>\lambda \) for \(t\ge T\). Noting \(g(t )< t\) and \(\lim_{t\to +\infty } g(t)=+\infty \), we can find \({T}'\ge T\) such that \(t>g(t)\ge T\) for \(t\ge T'\). Since the function \(x(t)\) is monotonically decreasing, we deduce that \(x(g(t))>x(t)>\lambda \) for \(t\ge T'\) and we have proved statement (i).

## 3 Asymptotic dichotomy

*t*approaches to infinity. Throughout this section we will suppose that one of the following conditions holds:

### Theorem 3.1

*Suppose that equation*(2.1)

*is non*-

*oscillatory and condition*(3.1)

*or*(3.2)

*holds*.

*We further assume that there exists a positive differentiable function*

*ρ*

*such that*

*In addition*,

*for*\(n\ge 2\),

*we further assume*\({p}'(t)\ge 0\)

*for*\(t\in [a,+\infty )\).

*Then any solution*\(y(t)\)

*of equation*(1.1)

*is oscillatory or converges to zero*,

*i*.

*e*., \(y(t)\to 0\)

*as*\(t\to +\infty \).

### Proof

Suppose that \(y(t)\) is a non-oscillatory solution of equation (1.1) on the interval \([T,+\infty )\), where \(T\ge a\). Without loss of generality, we may assume \(y(t)>0\) and \(y(g(t))>0\) for \(t\ge t_{0} \ge T\). By Lemma 2.1, there exists \(t_{1} \) such that \(y^{(n)}(t)>0\) or \(y^{(n)}(t)<0\) for \(t\ge t_{1} \ge t_{0} \).

*n*with \(n+2\) and use Lemma 2.3 to conclude \(l=n+1\). Since \(\lim_{t \to +\infty } g(t)= + \infty \) by our assumption, we may find \(t_{3} \ge t_{2} \) such that, for \(t\ge t_{3} \), we have \(g ( t )\ge t_{2} \). Hence, for \(t\ge t_{3} \), we obtain

*t*on both sides of the above inequality, we deduce that, for \(t\ge t_{3} \),

Therefore we know \(y^{(n)}(t)<0\) for \(t\ge t_{1} \). Noting that \(y(t)>0\) and \(y(g(t))>0\), we consider \(y^{(n+1)}(t)\) is either eventually negative or eventually positive.

First we prove Theorem 3.1 under condition (3.1). We now exclude the case that \(y^{(n+1)}(t)\) is eventually negative. In fact, if \(y^{(n)}(t)<0\) for \(t\ge t_{1} \). By using Lemma 2.4, we obtain \(\lim_{t\to +\infty } y^{ ( {n-1} )}(t)=- \infty \). Similarly, by induction on *n*, we deduce that \(y(t)\to - \infty \) as \(t\to +\infty \), which is a contradiction with \(y(t)>0\). Hence the function \(y^{ ( {n+1} )}(t)\) is eventually positive, and hence there exists \(t_{4} \ge t_{1} \) such that \(y^{ ( {n+1} )}(t)>0\) for \(t\ge t_{4} \).

We prove \(y(t)\to 0\) as \(t\to +\infty \) by using a contradiction argument. Otherwise, we assume \(\lim_{t\to +\infty } y(t)\ne 0\). We divide the proof into two cases with respect to *n*.

*s*to

*t*on both sides of (3.6), we get

*v*to

*t*on both sides of inequality (3.7) to get

*t*on both sides of the above inequality, we obtain

*t*, which leads to a contradiction with \(y(t)>0\).

We now prove for the case \(n\ge 2\). Then we know \(y^{ ( n )}(t)<0\), \(y(t)>0\) for \(t\ge t_{1} \) and \(\lim_{t\to +\infty } y(t)\ne 0\). By Lemma 2.5, we know there exist \(\lambda _{1} >0\) and \(t_{6} \ge t_{1} \) such that \(0< y^{ ( {n-1} )} ( t )<\lambda _{1} <y ( t )\), \(y ( {g ( t )} )>\lambda _{1} \) for \(t\ge t_{6} \).

*v*to

*t*on both sides of (3.9), we obtain

*t*on both sides of the above inequality to get

*t*, which also leads to a contradiction.

If \(y^{ ( {n+1} )}(t)\) does not change sign, then \(y^{ ( {n+1} )}(t)\) will not be eventually positive or eventually negative, which contradicts with \(y^{ ( n )}(t)<0\).

In the following, we prove Theorem 3.1 under condition (3.2). Then we know \(y^{ ( n )}(t)<0\) for \(t\ge t_{1} \). First we consider the case \(n=1\). By Lemma 2.5, we know \(\mu _{2} >0\) and there exists \(t_{7} \ge t_{1} \) such that \(y ( {g ( t )} )>y ( t )> \mu _{2} \) for \(t>t_{7}\).

*t*on both sides of equation (1.1) to get

Consider the case that \(n\ge 2\). By Lemma 2.5, there exist \(\lambda _{2} >0\) and \(t_{8} \ge t_{1} \) such that \(0< y^{ ( {n-1} )} ( t )<\lambda _{2} <y ( t )\) and \(y ( {g ( t )} )>\lambda _{2} \) for \(t\ge t _{8} \).

*t*on both sides of equation (1.1) to get

In the following, we prove a new asymptotic dichotomy for equation (1.1) under the so-called Philos-type integral averaging conditions. Following the literature [19], we first introduce a class of functions ℜ. We define two sets \(D_{0} =\{(t,s): t>s \ge T \}\) and \(D=\{(t,s):t\ge s \ge T\}\).

- (i)
\(H(t,t)=0\) for \(t\ge T\) and \(H(t,s)>0\) for \((t,s)\in D_{0} \);

- (ii)
*H*has a continuous and non-positive partial derivative on \(D_{0}\) with respect to the second variable such that \(- \frac{{\partial H ( {t, s} )} }{{\partial s}}=h(t,s) \sqrt[2]{H(t,s)}\) for \((t,s)\in D_{0}\);

*H*is said to belong to ℜ. □

### Theorem 3.2

*Suppose that equation*(2.1)

*is non*-

*oscillatory and condition*(3.1)

*or*(3.2)

*holds*.

*We further suppose that there exist two functions*\(H\in \Re \)

*and*\(0<\rho \in C^{1} ( { [ {T,\infty } )} )\)

*satisfying*

*where*

*For the case*\(n\ge 2\),

*we further assume*\({p}' ( t ) \ge 0\)

*for*\(t\in [ {a,+\infty } )\).

*Then any solution*\(y ( t )\)

*of equation*(1.1)

*is oscillatory or*\(y(t)\to 0\)

*as*\(t\to +\infty \).

### Proof

Suppose that \(y ( t )\) is a non-oscillatory solution of equation (1.1) on the interval \([T,+\infty )\), where \(T\ge a\). Without loss of generality we may assume \(y(t)>0\) and \(y(g(t))>0\) for \(t\ge t_{0} \ge T\). By Lemma 2.1, we deduce that \(y^{ ( n )}(t)>0\) or \(y^{ ( n )}(t)<0\) for \(t\ge t_{1} \ge t_{0} \).

If \(y^{ ( n )}(t)>0\) for \(t\ge t_{1} \), then by Lemma 2.2, we know \(y^{ ( {n+1} )} ( t )>0\).

*n*with \(n+2\) and use Lemma 2.3 to conclude that \(l=n+1\). Since \(\lim_{t\to +\infty }g(t)=+\infty \), then we may find \(t_{2} \ge t_{1} \) such that \(g(t)\ge t_{1} \) for \(t\ge t_{2} \). Therefore a relation similar to (3.5) holds for \(t\ge t_{2} \), i.e.,

### Theorem 3.3

*Suppose that equation*(2.1)

*is non*-

*oscillatory and condition*(3.1)

*or*(3.2)

*holds*.

*Assume further that there exist two functions*\(H\in \Re \)

*and*\(0< \rho \in C^{1}({[ {T,\infty })})\)

*such that*

*and*

*hold for all*

*T*,

*where*\(W ( t )\)

*is defined as*(3.17)

*in Theorem*3.2.

*In addition*,

*there exists*\(\varPsi \in C ({ [ {a,\infty } ),\mathbb{R}} )\)

*such that*

*and*

*where*\(\varPsi _{+} ( t )=\max \{ {\varPsi (t), 0} \}\).

*Then any solution*\(y(t)\)*of equation* (1.1) *is oscillatory or satisfies*\(y ( t )\to 0\)*as*\(t\to +\infty \).

### Proof

Suppose that \(y ( t )\) is a non-oscillatory solution of equation (1.1) on the interval \([T,+\infty )\), where \(T\ge a\). Without loss of generality, we may assume that \(y(t)>0\) and \(y(g(t))>0\) hold for \(t\ge t_{0} \ge T\). By Lemma 2.1, we deduce that \(y^{ ( n )}(t)>0\) or \(y^{ ( n )}(t)<0\) for \(t\ge t_{1} \ge t_{0} \).

*u*and

*v*as follows:

*χ*such that

*δ*be an arbitrary positive number. By (3.32), we know that there exists \(t_{2} \ge t_{1} \) such that, for \(t\ge t_{2}\),

*δ*, we conclude that

*M*.

*N*such that, for \(n\ge N\),

*n*. By the above inequality, we get

The case for \(y^{ ( n )}(t)<0\) is similar to the proof of Theorem 3.1, and we omit the details here. We thus have completed the proof of Theorem 3.3. □

## 4 Some applications of the asymptotic dichotomy

In this section, we give some examples to illustrate the applications of the asymptotic dichotomy proved in the previous section.

### Example 4.1

Hence condition (3.3) in Theorem 3.1 holds, and we can use Theorem 3.1 to conclude that all the solutions of equation (4.1) are oscillatory or approaching to zero as \(t\to +\infty \). In fact, we can verify that \(y ( t )=-\cos t\) is a solution of equation (4.1) which is oscillatory.

### Example 4.2

## Notes

### Acknowledgements

The authors would like to thank the referees for their valuable comments and suggestions which improve the manuscript.

### Availability of data and materials

All data generated or analysed during this study are included in this published article.

### Authors’ contributions

All authors contributed equally to this work. All authors read and approved the final manuscript.

### Funding

The first author is supported by the NSF of Guangdong Province of China (No. 2016A030307008) and the second author is supported by the NSF of China (No. 11771101), the major research program of Colleges and Universities in Guangdong Province (No. 2017KZDXM054), Guangzhou science and technology project (No. 201805010001) and Guangdong science and technology project (No. 2016B090927009).

### Competing interests

The authors declare that they have no competing interests.

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