Necessary and sufficient conditions for oscillation of the solutions of even order differential equations\(^{\bullet }\)

Cheng Jin-Fa\(^{\S }\) Chu Yu-Ming\(^{\ast }\)

December 15, 2010.

\(^\S \) School of Mathematical Sciences, Xiamen University, Siming South Str. no. 422, 361005, Siming District, Xiamen, China, e-mail: jfcheng@xmu.edu.cn.

\(^\ast \)Department of Mathematics, Huzhou Teachers College, Xueshi Str. no. 1, 313000, Huzhou, China, e-mail: chuyuming2005@yahoo.com.cn.

\(^{\bullet }\)The work of the second author has been supported by the Natural Science Foundation of China (Grant No. 11071069) and the Innovation Team Foundation of the Department of Education of Zhejiang Province (Grant No. T200924).

In this paper, we establish several necessary and sufficient conditions for oscillation of the solutions of the following even order differential equation

\[ x^{(n)}(t) + q(t)x^\gamma (t) = 0, \quad \mbox{$n$ is even}, \]

where \( q(t) \in C([t_0 ,\infty ),{\mathbb R}^+ )\) and \(\gamma \) is the quotient of odd positive integers.

MSC. 34K11.

Keywords. Oscillation, nonoscillatory solution, even order differential equation.

1 Introduction

Considering the \(n\)-order differential equation

\begin{equation} x^{(n)}(t) + q(t)x^\gamma (t) = 0,\quad \mbox{ $n$ is even}, \end{equation}
1

where \( q(t) \in C([t_0 ,\infty ),{\mathbb R}^+ )\) and \(\gamma \) is the quotient of odd positive integers.

In the recent past, the asymptotic and oscillatory properties of the solutions of \(n\)-order differential equations have been researched by many authors (see [1–3, 7–9]).

A solution of Eq.(1) is said to be oscillatory if it has arbitrarily large zeros. Otherwise, the solution is said to be nonoscillatory.

We say that Eq.(1) is strictly superlinear if \(\gamma {\gt}1\), strictly sublinear if \(0{\lt}\gamma {\lt}1\), and linear if \(\gamma =1\).

In particular, if \(n=2\), then Eq.(1) reduced to

\begin{equation} x''(t) + q(t)x^\gamma (t) = 0, \end{equation}
2

Eq.(2) is the well-known Emden-Fowler equation (see [10–12]).

Many remarkable results have been established for the oscillation of solutions of the second and higher order functional differential equations. For example, the following well-known Theorems A-C are presented in [4–6].

Theorem A (see [4, 6]). If \(\gamma {\gt}0\), then Eq.(2) has a bounded nonoscillatory solution if and only if

\[ \int _{t_0 }^\infty {sq(s){\rm d}s {\lt} \infty .} \]

Theorem B (see [4, 5]). If \(\gamma {\gt}1\), then all solutions of Eq.(2) are oscillatory if and only if

\[ \int _{t_0 }^\infty {sq(s){\rm d}s = \infty .} \]

Theorem C (see [6]). If \(0{\lt} \gamma {\lt}1\), then Eq.(2) is oscillatory if and only if

\[ \int _{t_0 }^\infty {s^{\gamma }q(s){\rm d}s = \infty .} \]

For Eq.(1) with \(\gamma =1\), the following Theorem D is presented in [7].

Theorem D. If \(\gamma =1\), then every bounded solution of Eq.(2) oscillates if and only if

\[ \int _{t_0 }^\infty {s^{n - 1}q(s){\rm d}s = \infty }. \]

Due to some obstacles of theoretical and technical character in handling with higher order nonlinear differential equation, and there are a few results which presented the necessary and sufficient conditions for the oscillatory behavior when \(\gamma \ne 1\). So there are a lot of problems worth to be considered further for the Eq.(1).

The main aim of this paper is to prove the following Theorem 1.1:

Theorem 1.1. If \(\gamma \ne 1\) is the quotient of odd positive integers and \(n\) is even, then the following statements are true:

\((a)\) If

\begin{equation} \int _{t_0 }^\infty {s^{n - 1} q(s){\rm d}s < \infty }, \end{equation}
3

then Eq.(1) has a bounded nonoscillatory solution;

\((b)\) If \(\gamma {\gt}1\), then every solution of Eq.(1) oscillates if and only if

\begin{equation} \int _{t_0 }^\infty {s^{n - 1} q(s){\rm d}s = \infty }; \end{equation}
4

\((c)\) If \(0{\lt}\gamma {\lt}1\), then every solution of Eq.(1) oscillates if and only if

\begin{equation} \int _{t_0 }^\infty {s^{(n - 1)\gamma } q(s){\rm d}s = \infty }. \end{equation}
5

We clearly see that Theorems A-C are the special case of our Theorem 1.1.

2 Proof of Theorem 1.1

In order to prove Theorem 1.1 we need the following Lemma 2.1.

Lemma 2.1 (see [1-2, 7]). Let \(x(t)\) be a positive and \(n\)-times differentiable function on \([t_0,\infty )\), and \(x^{(n)}(t)\) be nonpositive and not identically zero on any subinterval \([t_1,\infty )\). Then there exist \(T \ge t_0\) and integer \(k \in {\{ 0,1,...,n-1\} }\), such that \(n+k\) is odd and

\((i)\) \(x^{(i)}(t) \ge 0\) for \(t \ge T, i=0,1,...,k-1;\)

\((ii)\) \((-1)^{i+k}x^{(i)}(t){\gt}0\) for \(i=k,k+1,...,n;\)

\((iii)\) \((t-T)|x^{(k-i)}(t)| \le (1+i)|x^{(k-i-1)}(t)|\) for \(t\ge T,i=0,1,...,k-1,k=1,...,n-1.\)

Proof of the Theorem 1.1.

\((a)\) Assume that (3) holds, we first prove that Eq.(1) has a nonoscillatory solution.

Observing that if \(x(t)\) satisfies the equation

\begin{equation} x(t) = 1 - \tfrac {1}{{(n - 1)!}}\int _t^\infty {(s - t)^{n - 1} q(s)x^\gamma (s){\rm d}s}, \end{equation}
6

then \(x(t)\) is a solution of Eq.(1). Therefore it suffices to show that Eq.(6) has bounded nonoscillatory solution. To this end, choose sufficient large \(t \ge T\) such that

\begin{equation} \max \left\{ \int \limits _t^\infty s^{n - 1} q(s){\rm d}s,2\gamma \int \limits _t^\infty s^{n - 1} q(s){\rm d}s \right\} < \tfrac {1}{2} (n - 1)! .\end{equation}
7

Next, we consider the functional set

\[ M = \{ x \in C([T,\infty ),{\mathbb R}):\tfrac {1}{2} \le x(t) \le 1\} \]

and define the operator \(S:M \to C([T,\infty ),{\mathbb R})\) as follows:

\begin{equation} Sx(t) = 1 - \tfrac {1}{{(n - 1)!}}\int _t^\infty {(s - t)^{n - 1} q(s)x^\gamma (s){\rm d}s}.\end{equation}
8

We clearly see that \( x(t) ^\gamma \le 1\) and

\[ (Sx)(t) \ge 1 - \tfrac {1}{{(n - 1)!}}\int \limits _{t }^\infty {(s - t)^{n - 1} q(s){\rm d}s \ge \tfrac {1}{2} \quad for \ t \ge T.} \]

Therefore, \((Sx)(t)\le 1\) and \( S:M \to M\). Now, we claim that \(S\) is a contraction on \(M\). In fact, let \(f(x)=x^\gamma \), then for \(x_1,x_2 \in (\tfrac {1}{2},1)\) one has

\[ |x_1^\gamma - x_2^\gamma | = |f'(\xi )||x_1^{} - x_2^{} |, \ \text{ where} \ \xi \in (\min \{ x_1 ,x_2 \} ,\max \{ x_1 ,x_2 \} ), \]

where

\begin{eqnarray*} |f’(\xi )|=|\gamma \xi ^{\gamma -1}| & \le & \left\{ \begin{array}{lll} \gamma , & \ if \ \gamma \ge 1 \, , & \\[ 4pt] 2\gamma , & \ if \ 0{\lt}\gamma {\lt}1 \, . & \end{array} \right. \end{eqnarray*}

Therefore

\[ |x_1^\gamma - x_2^\gamma | \le 2\gamma |x_1^{} - x_2^{} |, \quad for \ x_1 ,x_2 \in (\tfrac {1}{2},1). \]

Let \(x,w \in M\), then for \(n\ge N\) one has

\begin{align*} |(Sx)(t) - (Sw)(t)| & \le \tfrac {1}{{(n - 1)!}}\int \limits _t^\infty {(s - t)^{n - 1} } q(s)|x^\gamma (s) - w^\gamma (s)|{\rm d}s\\ & \le \tfrac {2\gamma }{(n-1)!} \int \limits _t^\infty {(s - t)^{n - 1} } q(s)|x(s) - w(s)|{\rm d}s\\ & \le \tfrac {2\gamma }{(n-1)!} ||x(s) - w(s)||\int \limits _t^\infty {(s - t)^{n - 1} } q(s){\rm d}s \le \tfrac {1}{2}||x - w||. \end{align*}

Hence

\begin{equation} ||Sx - Sw|| \le \tfrac {1}{2}||x - w|| \end{equation}
9

and \(S\) is a contraction on \(M\). The (unique) fixed point of \(T\) is the desired bounded, nonoscillatory solution of Eq.(1).

\((b)\) Sufficiency. Assume that \(\gamma {\gt}1\) and \( \int _{t_0 }^\infty {s^{n - 1} q(s){\rm d}s = \infty }\), we prove that every solution of Eq.(1) oscillates. Otherwise, Eq.(1) has a nonoscillatory solution \(x(t)\). Without loss of generality, we assume that \(x(t){\gt}0\) for \(t\ge t_0\). Then Lemma 2.1 implies that there exist odd integer \(k \in {\{ 1,...,n-1\} }\) and \(T_k \ge t_0\) such that

\begin{equation} x^{(i)}(t)>0, \quad for \quad t \ge T_k,0 \le i \le k,\end{equation}
10

\begin{equation} (-1)^{i+k}x^{(i)}(t)>0, \quad for \quad t \ge T_k, k \le i \le n.\end{equation}
11

The proof is divided into two cases.

Case 1 \(k=1\). That is

\begin{equation} x'(t)>0,x''(t)<0,x^{(3)}(t)>0,...,x^{n}(t)<0.\end{equation}
12

From (10) and (11) together with the Taylor expansion we get

\begin{equation} x'(t) = \sum \limits _{j = 0}^{n - 2} {\tfrac {{( - 1)^j }}{{j!}}} x^{(1 + j)} (\tau )(\tau - t)^j + \tfrac {{( - 1)^{n - 1} }}{{(n - 2)!}}\int \limits _t^\tau {(s - t)^{n - 2} x^{(n)} (s){\rm d}s.} \end{equation}
13

Using (12) we have

\begin{equation} x'(t) > \int \limits _t^\tau {\tfrac {{(s - t)^{n - 2} }}{{(n - 2)!}}} q(s)x^\gamma (s){\rm d}s, \end{equation}
14

which implies

\begin{equation} x'(t) > \int \limits _t^\infty {\tfrac {{(s - t)^{n - 2} }}{{(n - 2)!}}} q(s)x^\gamma (s){\rm d}s > \int \limits _t^\infty {\tfrac {{(s - t)^{n - 2} }}{{(n - 2)!}}} q(s){\rm d}sx^\gamma (t). \end{equation}
15

From inequality

\begin{align*} \int _T^t (u - s)^{n - k - 1} {\rm d}s & = - \tfrac {{(u - s)^{u - k} }}{{n - k}}|_T^t = \tfrac {1}{{n - k}}[(u - T)^{n - k} - (u - t)^{n - k} ]\\ & \ge \tfrac {1}{{n - k}}(t - T)(u - T)^{n - k - 1} \end{align*}

we obtain

\begin{align*} \int \limits _{T_{} }^t {\tfrac {{x’(s)}}{{x^\gamma (s)}}} {\rm d}s& {\gt} \int \limits _{T_{} }^t {{\rm d}s\int \limits _s^\infty {} \tfrac {{(u - s)^{n - 2} }}{{(n - 2)!}}q(u)} {\rm d}u \\ & = \int \limits _{T_{} }^t {q(u){\rm d}u\int \limits _T^u {\tfrac {{(u - s)^{n - 2} }}{{(n - 2)!}}} } {\rm d}s + \int \limits _t^\infty {q(u){\rm d}u} \int \limits _T^t {\tfrac {{(u - s)^{n - 2} }}{{(n - 2)!}}} {\rm d}s\\ & \ge \int \limits _{T_{} }^t {\tfrac {{(u - T)^{n - 1} }}{{(n - 1)!}}q(u)} {\rm d}u + (t - T)\int \limits _{t_{} }^\infty {\tfrac {{(u - T)^{n - 2} }}{{(n - 1)!}}q(u)} {\rm d}u. \end{align*}

Therefore

\begin{equation} \int \limits _{T_{} }^t {\tfrac {{(u - T)^{n - 1} }}{{(n - 1)!}}q(u)}{\rm d}u< \int \limits _{T_{} }^t {\tfrac {{x’(s)}}{{x^\gamma (s)}}} {\rm d}s \end{equation}
16

or

\begin{equation} \int \limits _{T_{} }^t {\tfrac {{(u - T)^{n - 1} }}{{(n - 1)!}}q(u)} {\rm d}u < \tfrac {{x^{1 - \gamma } (t)}}{{\gamma - 1}} < \infty , \end{equation}
17

which contradicts with

\[ \int \limits _{T_{} }^\infty {u^{n - 1} q(u)} {\rm d}u = \infty . \]

Case 2 \(k{\gt}1\). It follows from \((iii)\) of Lemma 2.1 that for \(t \ge T_k\),

\begin{equation} x(t) \ge \tfrac {{(t - T_k )^{k - 1} }}{{k!}}x^{(k - 1)} (t).\end{equation}
18

For sufficient large \(t\), we have

\[ x^\gamma (t) \ge \tfrac {{(t - T_k )^{(k - 1)\gamma } }}{{(k!)^\gamma }}(x^{(k - 1)} (t))^\gamma {\gt} \tfrac {{(t - T_k )^{k - 1} }}{{(k!)^\gamma }}(x^{(k - 1)} (t))^\gamma ,\gamma {\gt} 1. \]

Let \(z(t)=x^{(k-1)}(t)\), then

\[ z(t){\gt}0,z'(t){\gt}0,z''(t){\lt}0,... \]

and so

\begin{equation} z^{(n - k + 1)} (t) + q(t)\tfrac {{(t - T_k )^{k - 1} }}{{(k!)^\gamma }}z^\gamma (t) < 0. \end{equation}
19

Making use of the same method as in the proof of case 1, we get

\[ \int _{t_0 }^\infty {s^{n - k}q(s)\tfrac {(s - T_k )^{k - 1}}{(k!)^{\gamma }}{\rm d}s {\lt} \infty } \]

or

\begin{equation} \int _{t_0 }^\infty {s^{n - 1} q(s)ds < \infty } , \end{equation}
20

which also contradicts with

\[ \int \limits _{t_0{} }^\infty {s^{n - 1} q(s)}{\rm d}s = \infty . \]

Conversely, we prove that (4) holds if every solution of Eq.(1) oscillates and \(\gamma {\gt}1\). Otherwise (3) holds, then from Theorem 1.1\((a)\) we get the contradiction that Eq.(1) has a nonoscillatory solution.

\((c)\) Sufficiency. For \(0{\lt}\gamma {\lt}1\), there are two cases as follows.

Case 1 \(k=1\). That is

\[ x(t){\gt}0,x'(t){\gt}0,x''(t){\lt}0,...,x^{n}(t){\lt}0. \]

Making use of the same method as Case 1 in Theorem 1.1\((b)\), we have

\begin{equation} x'(t) > \int \limits _t^\infty {\tfrac {{(s - t)^{n - 2} }}{{(n - 2)!}}} q(s)x^\gamma (s){\rm d}s. \end{equation}
21

Integrating (21) from \(T\) to \(t\) yields

\begin{align*} x(t)& {\gt} x(t) - x(T)\\ & {\gt} \int \limits _T^t {\tfrac {{(u - T)^{n - 1} }}{{(n - 1)!}}} q(u)x^\gamma (u){\rm d}u + (t - T)\int \limits _t^\infty {\tfrac {{(u - T)^{n - 2} }}{{(n - 1)!}}} q(u)x^\gamma (u){\rm d}u\\ & {\gt} (t - T)\int \limits _t^\infty {\tfrac {{(u - T)^{n - 2} }}{{(n - 1)!}}} q(u)x^\gamma (u){\rm d}u \end{align*}

or

\begin{equation} \tfrac {{x(t)}}{{t - T}} > \int \limits _t^\infty {\tfrac {{(u - T)^{n - 2} }}{{(n - 1)!}}} q(u)x^\gamma (u){\rm d}u. \end{equation}
22

Let

\begin{equation} z(t) = \int \limits _t^\infty {\tfrac {{(u - T)^{n - 2} }}{{(n - 1)!}}} q(u)x^\gamma (u){\rm d}u, \end{equation}
23

then \(z'(t){\lt}0,0{\lt}z(t){\lt} \tfrac {x(t)}{t-T}\) and

\begin{equation} z'(t) = - \tfrac {{(t - T)^{n - 2} }}{{(n - 1)!}}q(t)x^\gamma (t) \le -\tfrac {{(t - T)^{n - 2 + \gamma } }}{{(n - 1)!}}q(t)z^\gamma (t), \end{equation}
24

\begin{equation} \tfrac {{z’(t)}}{{z^\gamma (t)}} \le - \tfrac {{(t - T)^{n - 2 + \gamma } }}{{(n - 1)!}}q(t) \end{equation}
25

for \(T_2{\gt}T\). Then we get

\begin{equation} \int \limits _{T_2 }^t {\tfrac {{z’(u)}}{{z^\gamma (u)}}}{\rm d}u \le - \int \limits _{T_2 }^t {\tfrac {{(u - T)^{n - 2 + \gamma } }}{{(n - 1)!}}q(u)}{\rm d}u, \end{equation}
26

\begin{equation} \tfrac {1}{{1 - \gamma }}[z^{1 - \gamma } (t) - z^{1 - \gamma } (T_2 )] \le - \tfrac {1}{{(n - 1)!}}\int \limits _{T_2 }^t {(u - T)^{n - 2 + \gamma } q(u){\rm d}u}. \end{equation}
27

Therefore

\begin{equation} \int \limits _{T_2 }^t {(u - T)^{n - 2 + \gamma } q(u){\rm d}u} < + \infty . \end{equation}
28

Inequality (28) and \((n-1)\gamma {\lt}n-2+\gamma \) leads to

\begin{equation} \int \limits _{T_2 }^t {(u - T)^{(n - 1)\gamma } q(u){\rm d}u} < + \infty , \end{equation}
29

which contradicts with the assumption.

Case 2 \(k{\gt}1\). That is

\[ x(t){\gt}0, x'(t){\gt}0,..., x^{(k-1)}(t){\gt}0, x^{(k)}(t){\gt}0, x^{(k+1)}(t){\lt}0,...,x^{(n)}(t){\lt}0. \]

Lemma 2.1 implies

\[ x(t) \ge \tfrac {{(t - T_k )^{k - 1} }}{{k!}}x^{(k - 1)} (t) \]

or

\begin{equation} x^\gamma (t) \ge \tfrac {{(t - T_k )^{(k - 1)\gamma } }}{{(k!)^\gamma }}[x^{(k - 1)} (t)]^\gamma . \end{equation}
30

Let \(z(t)=x^{(k-1)}(t)\), then \(z(t){\gt}0,z'(t){\gt}0,z''(t){\lt}0,...,z^{(n - k + 1)}{\lt}0\) and

\begin{equation} z^{(n - k + 1)} (t) + q(t)\tfrac {{(t - T_k )^{(k - 1)\gamma } }}{{(k!)^\gamma }}z^\gamma (t) < 0, \end{equation}
31

where \(n-k+1\) is also even. Making use of the same method as in Case 1, we conclude that

\begin{equation} \int \limits _{t_0 }^\infty {s^{(n - k)\gamma } q(s)}\tfrac {(s - T_k )^{(k - 1)\gamma }}{(k!)^{\gamma }} {\rm d}s < + \infty \end{equation}
32

or

\begin{equation} \int \limits _{t_0 }^\infty {s^{(n - 1)\gamma } q(s)} {\rm d}s < + \infty , \end{equation}
33

which also contradicts with the assumption.

Necessity. For \(0{\lt}\gamma {\lt}1\) and (5) holds, we prove that Eq.(1) has a nonoscillatory solution. Otherwise, from (33) we know that there exists \(t \ge T\) such that

\begin{equation} \int \limits _t^\infty {s^{(n - 1)\gamma } } q(s){\rm d}s \le \tfrac {1}{2}. \end{equation}
34

Let \(M\) be a set defined by

\[ M=\{ x \in C([T, \infty ),R):\tfrac {1}{2(n-1)!}(t-T)^{n-1} \le x(t) \le \tfrac {1}{(n-1)!}(t-T)^{n-1},t \ge T\} \]

and the mapping \(T\) on \(M\) defined by

\begin{equation} Sx(t)=\int \limits _T^t {{\rm d}s_1 \int \limits _T^{s_1 } {{\rm d}s_2 \cdots } } \int \limits _T^{s_{n - 2} } {[\tfrac {1}{2}} + \int \limits _{s_{n - 1} }^\infty {q(u)x^\gamma (u){\rm d}u]{\rm d}s_{n - 1} }. \end{equation}
35

Then \((Sx)(t) \ge \tfrac {1}{2(n-1)!}(t-T)^{n-1}\) for \(x(t) \in M\) and \(t \ge T\). Moreover, from the definition of the operator \(S\) we get \((Sx)(t) \le \tfrac {1}{(n-1)!}(t-T)^{n-1}.\) Therefore, \(TM \subseteq M\).

Next, we define the function \(u_n:[T,\infty ) \to {\mathbb R}\) as follows

\begin{equation} u_n=(Su_{n-1})(t), \quad n \in {\mathbb N}\end{equation}
36

and

\[ u_0(t)=\tfrac {1}{2(n-1)!}(t-T)^{n-1}, \quad t \ge T. \]

A straightforward verification leads to

\[ \tfrac {1}{2(n-1)!}(t-T)^{n-1} \le u_{n-1}(t) \le u_n(t) \le \tfrac {1}{(n-1)!}(t-T)^{n-1}, \quad t \ge T. \]

Therefore, there exists the limit \( \mathop{\lim }\limits _{n \to \infty } u_n (t) = u(t)\) for \(t \ge T.\) It follows from the Lebesgue convergence theorem that \(u \in M\) and \(u(t)=(Su)(t).\) It is easy to see that \(u(t)\) is the solution of the Eq.(1).

Acknowledgement

The authors wish to thank the anonymous referees for their very careful reading of the manuscript and fruitful comments and suggestions.

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