Oscillatory behavior of second-order nonlinear neutral differential equations

Abstract

We shall consider a class of second-order nonlinear neutral differential equations. Some new oscillation criteria are established by using the Riccati transformation technique. One example is given to show the applicability of the main results.

1 Introduction

In this paper, we study the oscillation of a class of second-order nonlinear differential equations,

$$ \bigl( r(t) \bigl(z'(t)\bigr)^{\alpha } \bigr)^{\prime } + f\bigl(t,x\bigl(\sigma (t)\bigr)\bigr) = 0,\quad t \ge t_{0} > 0, $$

(1)

where $z(t) = x(t) - p(t)x(\tau (t))$, $\alpha > 0$, and α is the ratio of two odd integers. The following assumptions are satisfied:

($H_{1}$):

$r,p \in C([t_{0},\infty ),R), r(t) > 0, 0 \le p(t) \le p_{0} < 1$.

($H_{2}$):

$\tau \in C([t_{0},\infty ),R), \tau (t) \le t, \lim_{t \to \infty } \tau (t) = \infty$.

($H_{3}$):

$\sigma \in C^{1}([t_{0},\infty ),R),\sigma (t) \le t, \sigma '(t) > 0, \lim_{t \to \infty } \sigma (t) = \infty$.

($H_{4}$):

$f \in C(R,R)$, $uf(t,u) > 0$ for all $u \ne 0$, and there exists a function $q(t) \in C([t_{0},\infty ], [0,\infty ))$ such that $\vert f(t,u) \vert \ge q(t) \vert u^{\alpha } \vert$.

Second-order and third-order differential equations are widely used in population dynamics, physics, technology and other fields. Many scholars have studied the oscillation of second-order differential equations [1–10]. Similarly, many scholars have studied the oscillation of third-order differential equations [11–14]. On this basis, this paper studies the second-order neutral differential Eq. (1), Some new oscillation criteria are established by using the Riccati transformation technique.

2 Lemmas

In order to establish the oscillation criterion of Eq. (1), we will give three lemmas.

Lemma 2.1

Assume that

$$ \int _{ t_{0}}^{ \infty } r^{ - \frac{1}{\alpha }} (t)\,dt = \infty $$

(2)

and$x(t)$is an eventually positive solution of Eq. (1). Then$z(t)$has the following two possible cases:

• (i) $z(t) > 0$, $z'(t) > 0$, $( r(t)(z'(t))^{\alpha } )^{\prime } \le 0$;

• (ii) $z(t) < 0$, $z'(t) > 0$, $( r(t)(z'(t))^{\alpha } )^{\prime } \le 0$.

Proof

Since $x(t)$ is an eventually positive solution of (1), there exists a $t_{1} \ge t_{0}$ such that $x(t) > 0$, for $t \ge t_{1}$. From (1), we have

$$ \bigl( r(t) \bigl(z'(t)\bigr)^{\alpha } \bigr)^{\prime } \le 0 $$

hence $r(t)(z'(t))^{\alpha }$ is decreasing function and of one sign, therefore $z'(t)$ is also of one sign, that is, there exists a $t_{2} \ge t_{1}$ such that, for $t \ge t_{2}$, $z'(t) > 0$ or $z'(t) < 0$.

If $z'(t) > 0$, we have (i) or (ii). Now, we prove that $z'(t) < 0$ will not happen.

If $z'(t) < 0$, we have

$$ r(t) \bigl( - z'(t)\bigr)^{\alpha } \ge r(t_{2}) \bigl( - z'(t_{2})\bigr)^{\alpha } = K \ge 0, $$

where $K = r(t_{2})( - z'(t_{2}))^{\alpha } \ge 0$, that is,

$$ z'(t) \le - k^{\frac{1}{\alpha }} r^{ - \frac{1}{\alpha }} (t). $$

Integrating this inequality from $t_{2}$ to t, we have

$$ z(t) \le z(t_{2}) - k^{\frac{1}{\alpha }} \int _{ t_{2}}^{ t} r^{ - \frac{1}{\alpha }} (s)\,ds $$

by condition (2), $\lim_{t \to \infty } z(t) = - \infty$. We will consider the following two cases.

Case 1. If $x(t)$ is unbounded, then there exists a sequence $\{ t_{m} \}$, such that $\lim_{m \to \infty } t_{m} = \infty$ and $\lim_{m \to \infty } x(t_{m}) = \infty$, here $x(t_{m}) = \max \{ x(s):t_{0} \le s \le t_{m} \}$. Hence, we have

$$\begin{aligned} x\bigl(\tau (t_{m})\bigr) &= \max \bigl\{ x(s): t{}_{0} \le s \le \tau (t_{m}) \bigr\} \\ &\le \max \bigl\{ x(s): t{}_{0} \le s \le t_{m} \bigr\} = x(t_{m}). \end{aligned}$$

We get

$$ z(t_{m}) = x(t_{m}) - p(t_{m})x\bigl(\tau (t_{m})\bigr)\ge \bigl[1 - p(t_{m})\bigr]x(t_{m}) > 0. $$

This contradicts $\lim_{t \to \infty } z(t) = - \infty$.

Case 2. If $x(t)$ is bounded, then $z(t)$ is bounded, this contradicts $\lim_{t \to \infty } z(t) = - \infty$.

Hence, $z(t)$ satisfies one of the cases (i) and (ii). □

Lemma 2.2

Assume that$x(t)$is a positive solution of Eq. (1) and$z(t)$satisfies case (i) of Lemma2.1, then

$$z(t) \ge R(t)r^{\frac{1}{\alpha }} (t)z'(t),\qquad \biggl( \frac{z(t)}{R(t)} \biggr)^{\prime } \le 0, $$

where$R(t) = \int _{ T}^{ t} r^{ - \frac{1}{\alpha }} (s)\,ds$, $T \ge t_{0}$.

Proof

For $t > T \ge t_{0}$, we have

$$z(t) = z(T) + \int _{ T}^{ t} \frac{r^{\frac{1}{\alpha }} (s)z'(s)}{r^{\frac{1}{\alpha }} (s)}\,ds\ge r^{\frac{1}{\alpha }} (t)z'(t) \int _{ T}^{ t} r^{ - \frac{1}{\alpha }} (s)\,ds = R(t)r^{\frac{1}{\alpha }} (t)z'(t). $$

Thus, we conclude that

$$\biggl( \frac{z(t)}{R(t)} \biggr)^{\prime } = \frac{z'(t)R(t) - z(t)R'(t)}{R^{2}(t)}\le \frac{z'(t)R(t) - R(t)r^{\frac{1}{\alpha }} (t)z'(t)r^{ - \frac{1}{\alpha }} (t)}{R^{2}(t)} = 0. $$

 □

Lemma 2.3

Assume that$x(t)$is an eventually positive solution of (1) and

$$ \mathop{\lim \sup}_{t \to \infty } \int _{ \tau ^{ - 1}(\sigma (t))}^{ t} \biggl( \frac{1}{r(s)} \int _{ s}^{ t} q(u)\,du \biggr)^{\frac{1}{\alpha }} \,ds > p_{0}. $$

(3)

Then the impossibility for$z(t)$satisfies case (ii) of Lemma2.1.

Proof

Assume that $z(t)$ satisfies case (ii) of Lemma 2.1, we have

$$ - z(t) = - x(t) + p(t)x\bigl(\tau (t)\bigr) < p(t)x\bigl(\tau (t)\bigr)\le p_{0}x\bigl(\tau (t)\bigr). $$

That is,

$$ x\bigl(\tau (t)\bigr) \ge - \frac{1}{p_{0}}z(t). $$

We deduce that

$$ x(t) \ge - \frac{1}{p_{0}}z\bigl(\tau ^{ - 1}(t)\bigr),\qquad x\bigl(\sigma (t)\bigr) \ge - \frac{1}{p_{0}}z\bigl(\tau ^{ - 1}\bigl(\sigma (t) \bigr)\bigr). $$

From (1) and ($H_{4}$), we have

$$ \bigl( r(t) \bigl(z'(t)\bigr)^{\alpha } \bigr)^{\prime } + q(t) \bigl(x\bigl(\sigma (t)\bigr)\bigr)^{\alpha } \le 0. $$

We get

$$ \bigl( r(t) \bigl(z'(t)\bigr)^{\alpha } \bigr)^{\prime } + q(t) \biggl( - \frac{1}{p_{0}}\biggr)^{\alpha } z^{\alpha } \bigl(\tau ^{ - 1}\bigl(\sigma (t)\bigr)\bigr) \le 0. $$

Integrating this inequality from s to t, we conclude that

$$ r(t) \bigl(z'(t)\bigr)^{\alpha } - r(s) \bigl(z'(s) \bigr)^{\alpha } - \frac{1}{p_{0}^{\alpha }} \int _{ s}^{ t} q(u)z^{\alpha } \bigl(\tau ^{ - 1}\bigl(\sigma (u)\bigr)\bigr)\,du \le 0. $$

That is,

$$ - z'(s) \le \frac{1}{p_{0}} \biggl( \frac{1}{r(s)} \int _{ s}^{ t} q(u)z^{\alpha } \bigl(\tau ^{ - 1}\bigl(\sigma (u)\bigr)\bigr)\,du \biggr)^{\frac{1}{\alpha }}. $$

Integrating this inequality from $\tau ^{ - 1}(\sigma (t))$ to t, we get

$$ z\bigl(\tau ^{ - 1}\bigl(\sigma (t)\bigr)\bigr) - z(t) \le \frac{1}{p_{0}}z\bigl(\tau ^{ - 1}\bigl(\sigma (t)\bigr)\bigr) \int _{ \tau ^{ - 1}(\sigma (t))}^{ t} \biggl( \frac{1}{r(s)} \int _{ s}^{ t} q(u)\,du \biggr)^{\frac{1}{\alpha }} \,ds. $$

Since $z(t) < 0$, we have

$$ \int _{ \tau ^{ - 1}(\sigma (t))}^{ t} \biggl( \frac{1}{r(s)} \int _{ s}^{ t} q(u)\,du \biggr)^{\frac{1}{\alpha }} \,ds \le p_{0}. $$

This contradicts (3). Thus the impossibility for $z(t)$ satisfies case (ii) of Lemma 2.1. □

3 Oscillation results

Theorem 3.1

Assume that (2) and (3) be satisfied. If there exists a positive function$\rho \in C^{1}([t_{0},\infty ),(0,\infty ))$, such that, for all sufficiently large$T \ge t_{0}$,

$$ \int _{ t_{0}}^{ \infty } \biggl[\rho (t)\bar{Q}(t) - \frac{r(t)(\rho '(t))^{\alpha + 1}}{(\alpha + 1)^{\alpha + 1}\rho ^{\alpha } (t)}\biggr]\,dt = \infty, $$

(4)

where$\bar{Q}(t) = Q(t)\frac{R^{\alpha } (\sigma (t))}{R^{\alpha } (t)}$), $Q(t) = q(t)[1 + \bar{p}(\sigma (t))]^{\alpha }$, $\bar{p}(t) = p(t)\frac{R(\tau (t))}{R(t)}$, then Eq. (1) is oscillatory.

Proof

Assume that $x(t) > 0$. From Lemma 2.1, $z(t)$ satisfies one of the cases (i) and (ii).

Case (i). Suppose that case (i) holds, from Lemma 2.2, we have

$$ \frac{z(t)}{R(t)} \le \frac{z(\tau (t))}{R(\tau (t))}. $$

That is,

$$ z\bigl(\tau (t)\bigr) \ge R\bigl(\tau (t)\bigr)\frac{z(t)}{R(t)}. $$

We get

$$ z(t) = x(t) - p(t)x\bigl(\tau (t)\bigr)\le x(t) - p(t)z\bigl(\tau (t)\bigr)\le x(t) - p(t)R\bigl(\tau (t)\bigr)\frac{z(t)}{R(t)}. $$

That is,

$$ x(t) \ge \biggl[1 + p(t)\frac{R(\tau (t))}{R(t)}\biggr]z(t) = \bigl[1 + \bar{p}(t) \bigr]z(t), $$

where $\bar{p}(t) = p(t)\frac{R(\tau (t))}{R(t)}$.

From (1), we conclude that

$$ \bigl( r(t) \bigl(z'(t)\bigr)^{\alpha } \bigr)^{\prime } + q(t)x^{\alpha } \bigl(\sigma (t)\bigr) \le 0. $$

Then we have

$$ \bigl( r(t) \bigl(z'(t)\bigr)^{\alpha } \bigr)^{\prime } + q(t)\bigl[1 + \bar{p}\bigl(\sigma (t)\bigr)\bigr]^{\alpha } z^{\alpha } \bigl(\sigma (t)\bigr) \le 0. $$

That is,

$$ \bigl( r(t) \bigl(z'(t)\bigr)^{\alpha } \bigr)^{\prime } \le - Q(t)z^{\alpha } \bigl(\sigma (t)\bigr), $$

(5)

where $Q(t) = q(t)[1 + \bar{p}(\sigma (t))]^{\alpha }$.

We define a function $w(t)$ of the generalized Riccati transformation by

$$ w(t) = \frac{\rho (t)r(t)(z'(t))^{\alpha }}{z^{\alpha } (t)}. $$

Then $w(t) > 0$, from Lemma 2.2, we have $\frac{z(\sigma (t))}{R(\sigma (t))} \ge \frac{z(t)}{R(t)}$, that is, $\frac{z(\sigma (t))}{z(t)} \ge \frac{R(\sigma (t))}{R(t)}$.

Using the inequality [2]

$$ Bu - Au^{\frac{\theta + 1}{\theta }} \le \frac{\theta ^{\theta }}{(\theta + 1)^{\theta + 1}}\frac{B^{\theta + 1}}{A^{\theta }},\quad \theta > 0, A > 0, B \in R, $$

we have

$$\begin{aligned} w'(t) &= \rho '(t)\frac{r(t)(z'(t))^{\alpha }}{z^{\alpha } (t)} + \rho (t) \frac{(r(t)(z'(t))^{\alpha } )'}{z^{\alpha } (t)} - \rho (t)\frac{\alpha r(t)(z'(t))^{\alpha + 1}}{z^{\alpha + 1}(t)} \\ &\le \frac{\rho '(t)}{\rho (t)}w(t) - \rho (t)Q(t)\frac{z^{\alpha } (\sigma (t))}{z^{\alpha } (t)} - \frac{\alpha }{(\rho (t)r(t))^{{1 / \alpha }}} w^{\frac{\alpha + 1}{\alpha }} (t) \\ &\le \frac{\rho '(t)}{\rho (t)}w(t) - \rho (t)Q(t)\frac{R^{\alpha } (\sigma (t))}{R^{\alpha } (t)} - \frac{\alpha }{(\rho (t)r(t))^{{1 / \alpha }}} w^{\frac{\alpha + 1}{\alpha }} (t) \\ &\le - \rho (t)\bar{Q}(t) + \frac{\rho '(t)}{\rho (t)}w(t) - \frac{\alpha }{(\rho (t)r(t))^{{1 / \alpha }}} w^{\frac{\alpha + 1}{\alpha }} (t) \\ &= - \rho (t)\bar{Q}(t) + \frac{r(t)(\rho '(t))^{\alpha + 1}}{(\alpha + 1)^{\alpha + 1}\rho ^{\alpha } (t)}, \end{aligned}$$

(6)

where $\bar{Q}(t) = Q(t)\frac{R^{\alpha } (\sigma (t))}{R^{\alpha } (t)}$).

Integrating this inequality from T to t, we have

$$ w(t) \le w(T) - \int _{ T}^{ t} \biggl( \rho (s)\bar{Q}(s) - \frac{r(s)(\rho '(s))^{\alpha + 1}}{(\alpha + 1)^{\alpha + 1}\rho ^{\alpha } (s)} \biggr)\,ds. $$

From (4), we get $\lim w(t)_{t \to \infty } = - \infty$, this contradicts $w(t) > 0$.

Case (ii). If $z(t)$ satisfies (ii), then due to Lemma 2.3, Eq. (1) is oscillatory. □

Theorem 3.2

Assume that (2) and (3) are satisfied. If there exists a positive function$\varphi \in C^{1}([t_{0},\infty ),(0,\infty ))$such that, for all sufficiently large$T \ge t_{0}$,

$$ \int _{ t_{0}}^{ \infty } \biggl[\bar{Q}(t) - \frac{\varphi ^{\alpha + 1}(t)}{r^{1 / \alpha } (t)} \biggr]\exp \biggl[(\alpha + 1) \int _{ T}^{ t} \frac{\varphi (s)}{r^{1 / \alpha } (s)}\,ds\biggr] = \infty, $$

(7)

then Eq. (1) is oscillatory.

Proof

We use the counter-evidence method, suppose we have a non-oscillatory solution $x(t)$ of Eq. (1), as above, suppose that $x(t)$ is a positive solution of (1), by using Lemma 2.1, $z(t)$ satisfies one of (i) and (ii), we discuss each of the two cases separately.

Case (i). Assume that $z(t)$ has property (i), we obtain (5). We define a function $V(t)$ of a generalized Riccati transformation by

$$ V(t) = \frac{r(t)(z'(t))^{\alpha }}{z^{\alpha } (t)}. $$

Then $V(t) > 0$, using the Yang inequality $\frac{1}{p}a^{p} + \frac{1}{q}b^{q} \ge ab$, $\frac{1}{p} + \frac{1}{q} = 1$, similar to (6), we have

$$\begin{aligned} V'(t)& = \frac{(r(t)(z'(t))^{\alpha } )'}{z^{\alpha } (t)} - \frac{\alpha r(t)(z'(t))^{\alpha + 1}}{z^{\alpha + 1}(t)} \\ &\le - \bar{Q}(t) - \frac{\alpha }{r^{1 / \alpha } (t)}V^{\frac{\alpha + 1}{\alpha }} (t) \\ &= - \bigl[\bar{Q}(t) - r^{ - \frac{1}{\alpha }} (t)\varphi ^{\alpha + 1}(t)\bigr] - ( \alpha + 1)r^{ - \frac{1}{\alpha }} (t)\biggl[\frac{1}{\alpha + 1}\varphi ^{\alpha + 1}(t) + \frac{\alpha }{\alpha + 1}V^{\frac{\alpha + 1}{\alpha }} (t)\biggr] \\ &= - \bigl[\bar{Q}(t) - r^{ - \frac{1}{\alpha }} (t)\varphi ^{\alpha + 1}(t)\bigr] - ( \alpha + 1)r^{ - \frac{1}{\alpha }} (t)\varphi (t)V(t). \end{aligned}$$

That is,

$$ V'(t) + (\alpha + 1)r^{ - \frac{1}{\alpha }} (t)\varphi (t)V(t) \le - \bigl[ \bar{Q}(t) - r^{ - \frac{1}{\alpha }} (t)\varphi ^{\alpha + 1}(t)\bigr]. $$

We get

$$\begin{aligned} &\bigl[V'(t) + (\alpha + 1)r^{ - \frac{1}{\alpha }} (t)\varphi (t)V(t)\bigr]\exp [(\alpha + 1) \int _{ T}^{ t} \frac{\varphi (s)}{r^{1 / \alpha } (s)}\,ds \\ &\quad \le - \bigl[\bar{Q}(t) - r^{ - \frac{1}{\alpha }} (t)\varphi ^{\alpha + 1}(t)\bigr]\exp [( \alpha + 1) \int _{ T}^{ t} \frac{\varphi (s)}{r^{1 / \alpha } (s)}\,ds. \end{aligned}$$

That is,

$$\begin{aligned} &\biggl( V(t) \cdot \exp \biggl[(\alpha + 1) \int _{ T}^{ t} r^{ - \frac{1}{\alpha }} (s)\varphi (s)\,ds\biggr] \biggr)^{\prime }\\ &\quad \le - \bigl[\bar{Q}(t) - r^{ - \frac{1}{\alpha }} (t)\varphi ^{\alpha + 1}(t)\bigr]\exp [(\alpha + 1) \int _{ T}^{ t} \frac{\varphi (s)}{r^{1 / \alpha } (s)}\,ds. \end{aligned}$$

Integrating this inequality from T to t, we get

$$\begin{aligned} 0 &\le V(t) \cdot \exp \biggl[(\alpha + 1) \int _{ T}^{ t} r^{ - \frac{1}{\alpha }} (s)\varphi (s)\,ds\biggr]\\ &\le V(T) - \int _{ T}^{ t} \biggl( \bigl[\bar{Q}(t) - r^{ - \frac{1}{\alpha }} (t)\varphi ^{\alpha + 1}(t)\bigr]\exp [(\alpha + 1) \int _{ T}^{ t} \frac{\varphi (s)}{r^{1 / \alpha } (s)}\,ds \biggr)\,dt. \end{aligned}$$

This contradicts (7).

Case (ii). If $z(t)$ satisfies (ii), then due to Lemma 2.3, Eq. (1) is oscillatory. □

Example

Consider the following equation:

$$ \bigl( \bigl( x(t) - px(t - 1)' \bigr)^{\frac{1}{3}} \bigr)^{\prime } + q_{0}x^{\frac{1}{3}}(t - 2) = 0. $$

(8)

Comparing Eq. (8) with Eq. (1), let $r(t) = 1$, $\alpha = \frac{1}{3}$, $\tau (t) = t - 1$, $\sigma (t) = t - 2$, $q(t) = q_{0} > 0$, $p(t) = p < 1$ is a positive constant. Choose $\rho (t) = t$, $\varphi (t) = 1$, we now verify (3):

$$ \mathop{\lim \sup}_{t \to \infty } \int _{ \tau ^{ - 1}(\sigma (t))}^{ t} \biggl( \frac{1}{r(s)} \int _{ s}^{ t} q(u)\,du \biggr)^{1 / \alpha } \,ds = \mathop{\lim \sup}_{t \to \infty } \int _{ t - 1}^{ t} q_{0}(t - s)^{3} \,ds = \frac{q_{0}}{4} > p_{0}. $$

Therefore, if $\frac{q_{0}}{4} > p_{0}$, obviously, the conditions of Theorem 3.1 and Theorem 3.2 are satisfied, then Eq. (8) is oscillatory.

Then the conditions of Theorem 3.1 and Theorem 3.2 are satisfied.