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Asymptotic dichotomy in a class of higher order nonlinear delay differential equations
Journal of Inequalities and Applications volume 2019, Article number: 2 (2019)
Abstract
Employing a generalized Riccati transformation and integral averaging technique, we show that all solutions of the higher order nonlinear delay differential equation
will converge to zero or oscillate, under some conditions listed in the theorems of the present paper. Several examples are also given to illustrate the applications of these results.
Introduction
The objective of this paper is to investigate the oscillation and asymptotic behavior of solutions to the following higher order nonlinear delay differential equation:
where 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.
A solution of equation (1.1) is called oscillatory if it has arbitrarily large zeroes, otherwise it is called nonoscillatory. Equation (1.1) is called oscillatory if all its solutions are oscillatory. For solutions of some nonlinear delay differential equations, some interesting phenomena will occur. For example, some solutions are oscillatory, while there may exist some other solutions which are not oscillatory but will converge to zero as time approaches to infinity. We take a thirdorder delay differential equation as example. The equation
has an oscillatory solution \(y_{1}(t)=\sin (t)\) and a nonoscillatory solution \(y_{2}(t)=e^{\lambda t}\), where \(\lambda <0\) is a root to the characteristic function \(\lambda ^{3}+2\lambda +e^{\frac{\pi }{2} \lambda }=0\). Another example is that the delay differential equation \(y'''+y(t\tau )=0\) for \(\tau >0\) is oscillatory if and only if \(\tau e>3\). However, the corresponding equation \(y'''+y=0\) has a nonoscillatory solution \(y_{1}(t)=e^{t}\) and two oscillatory solutions \(y_{2}(t)=e^{\frac{t}{2}}\cos (\frac{\sqrt{3}t}{2})\) and \(y_{3}(t)=e ^{\frac{t}{2}}\sin (\frac{\sqrt{3}t}{2})\). So for higherorder delay differential equation, people naturally ask the following question: under what conditions does the equation have oscillatory or asymptotic behavior? This is the topic we will study in this paper.
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 higherorder nonlinear delay differential equations, to list all of which is almost impossible. We just list some studies relating to our work below.
In 1971 and 1977, Ladas [12] and Magfoud [16] studied the oscillation of solutions to the equation
where \(0< g(t)< t\), \(g(t)\to \infty \) as \(t \to +\infty \) and \(a(t)>0\).
In 2008, the authors studied in [2] the following 2norder nonlinear functional differential equation:
where α is the ratio of two positive odd integers. The oscillation theorems established in that paper extend a number of existing results.
In 2014, the oscillation and asymptotic behavior of solutions to the following nonlinear delay differential equation were studied in [15]:
In that paper, he used a generalized Riccati transformation and an integral averaging technique to study the sufficient conditions for the oscillations of differential equations. The goal of the present paper is to use a generalized method to study the oscillation and asymptotic behavior of solutions to the nonlinear delay differential equation (1.1). We need to carry out much more delicate analysis to overcome the difficulties in the proof. For the special case \(n=1\), equation (1.1) reduces to the form
which is exactly the main equation studied in [22] with \(r_{1} (t )=1\) and \(r_{2} ( t )=1\) as a special case. The third order differential equations arise in the study of entropyflow phenomenon, problems of hydrodynamics, threelayer beams and so forth, see the monograph [7] and the references cited therein. It is interesting to study the oscillation and asymptotic behavior of general higher order differential equation (1.1), and we give generalizations of the former studies in [22].
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.
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 nonoscillatory and suppose that \(y ( t )\) is a nonoscillatory 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
We use the contradiction argument to prove this lemma. Suppose that \(y ( t )\) is a nonoscillatory solution of equation (1.1). Without loss of generality, we may assume \(y(t)>0\) and \(y(g(t))>0\) for \(t\ge t_{0} \ge T\). It is easy to see that \(x(t)=y^{ ( n )}(t)\) is the solution of the following second order delay differential equation:
Suppose that \(z ( t )\) is a nonoscillatory solution of equation (2.1). Without loss of generality, we may assume \(z(t)>0\) for \(t\ge t_{0} \). Suppose that \(x ( t )\) is an oscillatory solution to equation (2.2), 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
i.e., we have
Integrating from a to b on both sides of equation (2.3), we obtain
which leads to a contradiction, and we thus complete the proof of Lemma 2.1. □
Lemma 2.2
Suppose that equation (2.1) is nonoscillatory, \(y ( t )\) is a nonoscillatory 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
Let \(y ( t )\) be a nonoscillatory solution to equation (1.1) such that \(y(t)y^{ ( n )}(t)>0\) for \(t\ge T\ge a\). Without loss of generality, we may assume \(y(t)>0\), \(y(g(t))>0\), and \(y^{ ( n )}(t)>0\) (\(t\ge t_{0} \ge T\)). Using equation (1.1), we know
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.
We use a contradiction argument to exclude the case that \(y^{ ( {n+1} )} ( t )\) is eventually negative. In fact, if \(y^{ ( {n+1} )} ( t )\) is eventually negative, without loss of generality, we may assume \(y^{ ( {n+1} )} ( t )<0\) for \(t\ge T\). By using the monotone decreasing of \(y^{ ( {n+1} )} ( t )\), we have
Integrating from T to t on both sides of inequality (2.5), we obtain
By letting \(t\to +\infty \) in the above inequality, we see \(y^{ ( n )} ( t )\to \infty \) and this leads to a contradiction. Therefore \(y(t)y^{ ( {n+1} )} ( t )\) is eventually positive and hence we have completed the proof of Lemma 2.2.
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 \(l\in \{ {1, 3,\ldots , n1} \}\) (respectively \(l\in \{ {0,2,4,\ldots , n1} \}\)) such that, for all sufficiently large t, \(x ( t ) x^{ ( j )} ( t )>0\) for \(j=0, 1,\ldots , l\) and \(( {1} )^{n+j1}x ( t ) x^{ ( j )} ( t )>0\) for \(j=l+1, l+2,\ldots , n1\). 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 \(0< x^{({n1})}(t )<\lambda <x(t )\), \(x({g(t )})> \lambda \) 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).
Now we suppose that \(n\ge 2\). Since \(x^{(n)}(t)<0\) for \(t\ge T\), we deduce that \(x^{({n1})}(t)\) is monotonically decreasing on the interval \([T,+\infty )\), and therefore \(x^{({n1})}(t)\) is eventually positive or eventually negative. We use a contradiction argument to exclude the case for eventual negativeness. In fact, if \(x^{({n1} )}(t)\) is eventually negative, by using the fact that \(x^{(n )}(t)<0\) for \(t\ge T\) and Lemma 2.4, we obtain \(\lim_{t\to +\infty } x ^{({n2})}(t)=\infty \). Similarly, by using the induction method, we know \(x(t)\to \infty \) as \(t\to +\infty \), which is a contradiction with \(x(t)>0\). Hence \(x^{({n1} )}(t)\) is eventually positive. Without loss of generality, we may assume \(x^{({n1} )}(t)>0\) for \(t\ge T\) and \(x^{({n1} )}(t)\) is monotonically decreasing on the interval \([T,+\infty )\). By using the monotone bounded theorem, we know
From Lemma 2.3 we know that \(x(t)\) is eventually strictly monotonous and \(x(t)>0\) for \(t\ge T\). Since \(\lim_{t\to +\infty } x(t) \ne 0\), we assume \(\lim_{t\to +\infty } x(t)=\lambda _{0} \), where \(\lambda _{0} >0\) or \(\lambda _{0} =+\infty \). We divide the proof of statement (ii) into two cases. We first deal with the case for \(\mu =0\). Let \(\lambda \in ({0, \lambda _{0} })\), then there exists \(T_{1} \ge T\) such that \(0< x^{({n1})}(t)<\lambda <x(t)\) for \(t\ge T_{1} \). Since \(\lim_{t\to +\infty } g(t )=+\infty \), then there exists \({T}'\ge T_{1} \) such that \(g(t )\ge T\) for \(t\ge {T}'\). Therefore we know \(x({g(t)})>\lambda \) and \(0< x^{({n1})}(t)<\lambda <x(t)\) for \(t\ge {T}'\), and hence we have proved that statement (ii) holds in this case. For the second case \(\lim_{t\to +\infty } x^{( {n1} )}(t)=\mu >0\), we repeatedly use Lemma 2.4 to deduce \(\lambda _{0} =+\infty \). If we let \(\lambda \in ({\mu ,+\infty })\), then there exists \({T}'\ge T\) such that \(0< x^{({n1})}(t )<\lambda <x(t)\) and \(x({g(t)})>\lambda \) for \(t\ge {T}'\). Statement (ii) also holds in this case, and we have completed the proof of Lemma 2.5. □
Asymptotic dichotomy
In this section we present some sufficient conditions which guarantee that every solution to equation (1.1) oscillates or converges to zero as t approaches to infinity. Throughout this section we will suppose that one of the following conditions holds:
or
where the integrand \(\mathit{Kq}(t)p'(t)\ge 0\) for \(t\in [a,+\infty )\) and does not vanish identically on any subinterval of \([a,+\infty )\).
Theorem 3.1
Suppose that equation (2.1) is nonoscillatory 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 nonoscillatory 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 \(t\ge t_{1} \ge t_{0} \).
If \(y^{(n)}(t)>0\) for \(t\ge t_{1} \), by using Lemma 2.2, we know that there exists \(t_{2} \ge t_{1} \) such that \(y^{({n+1})}(t)>0\) for \(t\ge t_{2} \). We now define a function
Since \(\rho (t)\) is positive, we know that the function \(\omega (t)>0\). Using equation (1.1), we know \(y^{(n+2)} ( t )<0\) for \(t\ge t_{2} \). Since \(y(t)>0\), \(y^{(n)}(t)>0\), \(y^{ ( {n+1} )} ( t )>0\), and \(y^{ ( n )} ( t )>0\) for \(t\ge t_{2} \), then we can replace 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
By using equation (1.1), (3.4), (3.5), and Lemma 2.3, we know that, for \(t\ge t_{3} \),
Integrating from \(t_{3} \) to t on both sides of the above inequality, we deduce that, for \(t\ge t_{3} \),
which is a contradiction with (3.3) valuing at \(T=t_{3} \).
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 \(\lim_{t\to +\infty } y^{ ( {n1} )}(t)= \infty \). 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.
First we prove for the special case \(n=1\). Then we have \({y}''(t)>0\), \({y}'(t)<0\), \(y(t)>0\) for \(t\ge t_{4} \). By Lemma 2.5, we know there exist \(\mu >0\) and \(t_{5} \ge t_{4} \) such that \(y ( {g ( t )} )>y ( t )>\mu \) for \(t\ge t _{5} \). By equation (1.1), we get
Integrating from s to t on both sides of (3.6), we get
for \(t\ge s\ge t_{5} \). Letting \(t\to +\infty \) in the above inequality, we obtain
We integrate from v to t on both sides of inequality (3.7) to get
Integrating from \(t_{5} \) to t on both sides of the above inequality, we obtain
Combining (3.1) and (3.8), we know \(y ( t )<0\) for sufficiently large 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 \(0< y^{ ( {n1} )} ( t )<\lambda _{1} <y ( t )\), \(y ( {g ( t )} )>\lambda _{1} \) for \(t\ge t_{6} \).
By equation (1.1), we deduce that, for \(t\ge s\ge t_{6} \),
i.e., for \(t\ge v\ge t_{6} \), we know
Integrating from v to t on both sides of (3.9), we obtain
We integrate from \(t_{6} \) to t on both sides of the above inequality to get
Combining (3.1) and (3.10), we deduce that \(y^{(n1)}(t)<0\) for sufficiently large 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 \(t>t_{7}\).
We integrate from \(t_{7} \) to t on both sides of equation (1.1) to get
Therefore we have
From the above inequality, we easily obtain
where \(H_{1}\) is a constant. Combining (3.2) and (3.11), we conclude that \({y}'' ( t )\) is eventually negative.
Consider the case that \(n\ge 2\). By Lemma 2.5, there exist \(\lambda _{2} >0\) and \(t_{8} \ge t_{1} \) such that \(0< y^{ ( {n1} )} ( t )<\lambda _{2} <y ( t )\) and \(y ( {g ( t )} )>\lambda _{2} \) for \(t\ge t _{8} \).
We integrate from \(t_{8} \) to t on both sides of equation (1.1) to get
i.e.,
where \(H_{2} \) is another constant. Combining (3.2) and (3.13), we know \(y^{ ( {n+1} )} ( t )\) is eventually negative. Noting that \(y^{ ( n )}(t)<0\) for \(t\ge t_{1} \), we use Lemma 2.4 to conclude that \(\lim_{t\to +\infty } y^{ ( {n1} )} ( t )= \infty \). Similarly, we can use the induction method to obtain \(\lim_{t\to +\infty } y ( t )=\infty \). This is a contradiction and we finish the proof of Theorem 3.1.
In the following, we prove a new asymptotic dichotomy for equation (1.1) under the socalled Philostype 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 \(D=\{(t,s):t\ge s \ge T\}\).
If a function \(H\in C(D,\mathbb{R})\) satisfies

(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 nonpositive 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 \((t,s)\in D_{0}\);
then H is said to belong to ℜ. □
Theorem 3.2
Suppose that equation (2.1) is nonoscillatory 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 nonoscillatory 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 \(y^{ ( {n+1} )} ( t )>0\).
We define \(\omega ( t )\) as in (3.4), i.e.,
for \(t\ge t_{1} \). It is easy to see that \(\omega ( t )>0\). Using equation (1.1), we get \(y^{ ( {n+2} )} ( t )<0\) for \(t\ge t_{1} \). Note that \(y(t)>0\), \(y^{ ( n )}(t)>0\), \(y^{ ( {n+1} )} ( t )>0\), and \(y^{ ( {n+2} )} ( t )<0\) for \(t\ge t_{1} \). We replace 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.,
Combining (1.1), (3.4), the above inequality, and Lemma 2.3, it follows that
Since
and
we carry out the estimates of inequality (3.18) and obtain
Therefore
where \(Q(t,s)=h(t,s) \sqrt[2]{H ( {t, s} )} \gamma (s )\).
From inequality (3.19), we deduce that
which contradicts with (3.14). The remaining proof is similar to the proof of Theorem 3.1, and we omit the details here. Therefore we have completed the proof of Theorem 3.2. □
Theorem 3.3
Suppose that equation (2.1) is nonoscillatory 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 nonoscillatory 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 \(t\ge t_{1} \ge t_{0} \).
If \(y^{(n)}(t)>0\) for \(t\ge t_{1} \). From Lemma 2.2, we know \(y^{ ( {n+1} )} ( t )>0\). We define \(\omega ( t )\), \(Q ( {t, s} )\), \(\gamma ( t )\), and \(W ( t )\) as in (3.4), (3.15), (3.16), and (3.17), respectively. The same as in the proof of (3.18) in Theorem 3.2, we know that inequality (3.18) remains true here. By (3.18), we know that
From inequality (3.25), it follows that
Combining (3.24) and (3.26), we deduce that
i.e.,
We define two functions u and v as follows:
and
Combining (3.22) and (3.27), we deduce that
Using (3.30), we claim and prove the following inequality:
We prove it by using a contradiction argument. If (3.31) were not true, then
By (3.21), there exists a positive constant χ such that
Let δ be an arbitrary positive number. By (3.32), we know that there exists \(t_{2} \ge t_{1} \) such that, for \(t\ge t_{2}\),
By (3.28), we know that, for \(t\ge t_{2}\),
By (3.21), we know that there exists \(t_{3} \ge t_{2} \) such that, for \(t\ge t_{3} \),
Combining (3.35) and (3.36), we know \(u ( t ) \ge \delta \) for \(t\ge t_{3} \). From the arbitrariness of δ, we conclude that
which is a contradiction with (3.30), and we have proved claim (3.31).
From (3.30), we can choose a sequence \(\{t_{n}\}_{n=1}^{ \infty } \subset (t_{0} ,\infty )\) satisfying \(\lim_{n\to +\infty } t _{n} = +\infty \) such that
for some positive number M.
Combining (3.28) and (3.32), we obtain
Combining (3.38) and (3.39), we get
By (3.38) and (3.39), we know there exists a positive integer N such that, for \(n\ge N\),
Combining (3.39), (3.40), and (3.41), we obtain the following:
On the other hand, by using the Schwarz inequality, we obtain
for any arbitrary integer n. By the above inequality, we get
Combining (3.42) and (3.43), we deduce that
By the arbitrariness of the sequence \(\{ {t_{n} } \}_{n=1} ^{\infty }\), it is obvious that (3.44) contradicts condition (3.22) in Theorem 3.3.
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. □
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
Consider the fourthorder delay differential equation
Here, comparing with equation (1.1), we see \(p(t)=\frac{1}{8t ^{2}}\), \(q(t)=1\frac{1}{8t^{2}}\), \(f ( u )=u\) with \(K=1\) and \(g(t)=t\pi \).
It is easy to verify that equation (4.1) satisfies condition (3.2) and that the equation \({z}''+\frac{1}{8t^{2}}z=0\) is nonoscillatory. We now take \(\rho =1\) and then obtain
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
Consider the fifthorder delay differential equation in the following:
where comparing with equation (1.1), we take \(p(t)=e^{2t+2}\), \(q(t)=\frac{1}{e}\), \(f ( u )=u ( {1+u^{2}} )\) with \(K=1\) and \(g(t)=t1\).
It is easy to verify that equation (4.2) satisfies condition (3.2) and the equation \({z}''+e^{2t+2}z=0\) is nonoscillatory. We choose \(H ( {t,s} )= ( {\frac{ ( {t2} ) ^{3}}{6}\frac{ ( {s2} )^{3}}{6}} )^{2}\) and \(\rho =1\) and verify that
Therefore condition (3.14) in Theorem 3.2 holds, and we use Theorem 3.2 to conclude that all the solutions of equation (4.2) are oscillatory or approaching to zero as \(t\to +\infty \). It is easy to verify that \(y ( t )=e^{t}\) is a solution of equation (4.2) which is approaching to zero as \(t\to +\infty \).
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The authors would like to thank the referees for their valuable comments and suggestions which improve the manuscript.
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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).
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Ye, Y., Liang, H. Asymptotic dichotomy in a class of higher order nonlinear delay differential equations. J Inequal Appl 2019, 2 (2019). https://doi.org/10.1186/s1366001819497
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DOI: https://doi.org/10.1186/s1366001819497
MSC
 34K11
 34K25
Keywords
 Asymptotic behavior
 delay differential equation
 higher order differential equation
 oscillation
 Schwarz inequality