Oscillation of forced second-order neutral delay differential equations

Abstract

The objective of this paper is to study oscillation of a forced second-order neutral differential equation. By using the generalized Riccati substitution and integral technique, a new sufficient condition is obtained which insures that all solutions to the studied equation are oscillatory. An illustrative example is included.

1 Introduction

In this paper, we are concerned with the oscillation of a forced second-order nonlinear neutral differential equation

$$ \bigl(r(t)\bigl[x(t)+P(t)x\bigl(\tau(t)\bigr)\bigr]' \bigr)'+\sum^{m}_{i=1}Q_{i}(t)f_{i} \bigl(x(t)\bigr) +\sum^{l}_{j=1}R_{j}(t)g_{j} \bigl(x\bigl(\tau(t)\bigr)\bigr)=F(t), $$

(1.1)

where $t\geq t_{0}>0$, $m\geq1$, and $l\geq1$ are integers. We suppose that the following assumptions are satisfied:

(A₁):

$r\in\mathrm{C}^{1}([t_{0}, \infty),(0, \infty))$, $P, Q_{i}, R_{j}\in\mathrm{C}([t_{0}, \infty),[0, \infty))$, $f_{i}, g_{j}\in \mathrm{C}(\mathbb{R},\mathbb{R})$, $yf_{i}(y)>0$, and $yg_{j}(y)>0$ for $y\neq0$, $i=1,2,\ldots,m$, and $j=1,2,\ldots,l$;

(A₂):

$\tau\in\mathrm{C}([t_{0}, \infty),\mathbb{R})$, $\tau (t)\leq t$, and $\lim_{t\rightarrow\infty}\tau(t)=\infty$;

(A₃):

there exist constants $\alpha_{i}>0$ and $\beta_{j}>0$ such that ${f_{i}(y)}/{y} \geq\alpha_{i}$ and ${g_{j}(y)}/{y} \geq\beta_{j}$ for $y\neq0$, $i=1,2,\ldots,m$, and $j=1,2,\ldots,l$;

(A₄):

for any $T\geq t_{0}$, there exist $T\leq s_{1}< t_{1}\leq s_{2}< t_{2}$ such that

$$F(t)\left \{ \textstyle\begin{array}{@{}l@{\quad}l} \leq0, & t\in[s_{1}, t_{1}],\\ \geq0, & t\in[s_{2}, t_{2}], \end{array}\displaystyle \right . $$

and

$$ \sum^{l}_{j=1} \beta_{j}R_{j}(t)\geq\sum^{m}_{i=1} \alpha_{i}Q_{i}(t)P(t),\quad t\in[s_{1}, t_{1}]\cup[s_{2}, t_{2}]. $$

(1.2)

Throughout the paper, we define

$$ z(t):=x(t)+P(t)x\bigl(\tau(t)\bigr). $$

(1.3)

By a solution of (1.1) we mean a function $x\in\mathrm{C}([T_{x} , \infty), \mathbb{R})$, $T_{x}\geq t_{0}$, which has the property $rz'\in\mathrm{C}^{1}([T_{x} , \infty), \mathbb{R})$ and satisfies (1.1) on $[T_{x} , \infty)$. We consider only those solutions x of (1.1) which satisfy condition $\sup\{|x(t)|: t\geq T\}>0$ for all $T\geq T_{x}$. We assume that (1.1) possesses such solutions. A solution of (1.1) is called oscillatory if it has arbitrarily large zeros on the interval $[T_{x}, \infty)$; otherwise, it is termed nonoscillatory.

As is well known, the study of qualitative theory of differential equations is of importance both in theory and applications. For instance, the problems of oscillatory behavior of neutral differential equations have a number of practical applications in the study of distributed networks containing lossless transmission lines which arise in high-speed computers where the lossless transmission lines are used to interconnect switching circuits. For some related contributions on oscillation of various classes of neutral differential equations, we refer the reader to [1–23] and the references cited therein.

In what follows, we provide some background details that motivated our study. El-Sayed [4] and Wong [19] investigated the second-order forced linear differential equation

$$\bigl(p(t)x'\bigr)'+q(t)x=f(t). $$

Zhang et al. [22] studied a second-order neutral differential equation

$$ \bigl(r(t)\bigl[x(t)+p(t)x(t-\tau)\bigr]' \bigr)'+Q_{1}(t)f\bigl(x(t)\bigr)+Q_{2}(t)g \bigl(x(t-\tau)\bigr)=H(t), $$

(1.4)

where $Q_{1}$ and $Q_{2}$ are nonnegative functions. Equation (1.4) is a special case of (1.1). In the sequel, using a generalized Riccati substitution which differs from those exploited in [4, 19, 22], a new oscillation criterion for (1.1) is presented. Furthermore, an illustrative example is provided.

2 Main results

Theorem 2.1

Assume that conditions (A₁)-(A₄) are satisfied and let $B_{k}=\lbrace u\in\mathrm{C}^{1}[s_{k}, t_{k}]: u(t)\not\equiv0, u(s_{k})=u(t_{k})=0\rbrace$, $k=1$, 2. If there exist functions $u\in B_{k}$, $\rho\in\mathrm{C}^{1}([t_{0}, \infty), (0, \infty))$, and $\sigma\in\mathrm{C}^{1}([t_{0}, \infty), \mathbb{R})$ such that, for $k=1$, 2,

$$\begin{aligned} &J_{k}(u, \rho, \sigma) \\ &\quad=\int_{s_{k}}^{t_{k}} \Biggl\{ \rho \Biggl[u^{2} \Biggl(\sum^{m}_{i=1} \alpha _{i}Q_{i}+r\sigma^{2}-(r \sigma)' \Biggr) -r \biggl(u'+\frac{u\rho'}{2\rho}+u\sigma \biggr)^{2} \Biggr] \Biggr\} (t) \,\mathrm{d}t>0, \end{aligned}$$

(2.1)

then every solution of (1.1) is oscillatory.

Proof

Suppose that x is a nonoscillatory solution of (1.1) which is eventually positive. Then z defined by (1.3) is also eventually positive. Using (A₄), for any $T\geq t_{0}$, there exist $t_{1}>s_{1}\geq T$ such that $F(t)\leq0$ for $t\in[s_{1}, t_{1}]$. From (A₃), (1.1), (1.2), and (1.3), we have

$$\begin{aligned} \bigl(rz'\bigr)'(t) =&F(t)-\sum ^{m}_{i=1}Q_{i}(t)f_{i} \bigl(x(t)\bigr)-\sum^{l}_{j=1}R_{j}(t)g_{j} \bigl(x\bigl(\tau(t)\bigr)\bigr) \\ \leq&-\sum^{m}_{i=1}\alpha_{i}Q_{i}(t)x(t)- \sum^{l}_{j=1}\beta_{j}R_{j}(t)x \bigl(\tau(t)\bigr) \\ \leq&- \Biggl[\sum^{m}_{i=1} \alpha_{i}Q_{i}(t)x(t)+\sum^{m}_{i=1} \alpha _{i}Q_{i}(t)P(t)x\bigl(\tau(t)\bigr) \Biggr] \\ =&-\sum^{m}_{i=1}\alpha_{i}Q_{i}(t)z(t). \end{aligned}$$

(2.2)

For $t\geq T$, we define a generalized Riccati substitution by

$$ V(t):=-\rho(t) \biggl[\frac{r(t)z'(t)}{z(t)}+r(t)\sigma(t) \biggr]. $$

(2.3)

Then we have

$$\begin{aligned} V' =&-\rho' \biggl(\frac{rz'}{z}+r \sigma \biggr)-\rho \biggl(\frac {rz'}{z}+r\sigma \biggr)' \\ =&\frac{\rho'}{\rho}V-\rho \biggl(\frac{rz'}{z} \biggr)'-\rho (r\sigma )' \\ =&\frac{\rho'}{\rho}V-\rho\frac{(rz')'}{z}+\rho\frac{r(z')^{2}}{z^{2}}-\rho (r\sigma )'. \end{aligned}$$

(2.4)

By virtue of (2.3), we obtain

$$ \biggl(\frac{z'}{z} \biggr)^{2}= \biggl( \frac{V}{-\rho r}-\sigma \biggr)^{2} = \biggl(\frac{V}{\rho r} \biggr)^{2}+\sigma^{2}+2\frac{V\sigma}{\rho r}. $$

(2.5)

For $t\in[s_{1}, t_{1}]$, substituting (2.2) and (2.5) into (2.4), we conclude that

$$\begin{aligned} V' =&\frac{\rho'}{\rho}V-\rho\frac{(rz')'}{z}+\rho r \biggl(\frac {V^{2}}{\rho^{2}r^{2}}+\sigma^{2} +2\frac{V\sigma}{\rho r} \biggr)-\rho (r\sigma )' \\ =&-\rho\frac{(rz')'}{z}+\rho \bigl[r\sigma^{2}-(r \sigma)' \bigr]+ \biggl(\frac{\rho'}{\rho} +2\sigma \biggr)V+ \frac{V^{2}}{\rho r} \\ \geq&\rho \Biggl[\sum^{m}_{i=1} \alpha_{i}Q_{i}+r\sigma^{2}-(r \sigma)' \Biggr]+ \biggl(\frac{\rho'}{\rho} +2\sigma \biggr)V+ \frac{V^{2}}{\rho r}. \end{aligned}$$

(2.6)

Let $u\in B_{1}$ be given as in the hypothesis. Multiplying (2.6) by $u^{2}$ and integrating the resulting inequality from $s_{1}$ to $t_{1}$, we have

$$\begin{aligned} \int_{s_{1}}^{t_{1}}u^{2}V' \,\mathrm{d}t\geq{}&\int_{s_{1}}^{t_{1}}u^{2}\rho \Biggl[\sum^{m}_{i=1}\alpha_{i}Q_{i}+r \sigma^{2}-(r\sigma)' \Biggr]\,\mathrm{d}t +\int _{s_{1}}^{t_{1}} \biggl(\frac{\rho'}{\rho}+2\sigma \biggr)Vu^{2}\,\mathrm{d}t \\ &{}+\int_{s_{1}}^{t_{1}} \frac{V^{2}}{\rho r}u^{2}\,\mathrm{d}t. \end{aligned}$$

(2.7)

Integrating (2.7) by parts and using the fact that $u(s_{1})=u(t_{1})=0$, we deduce that

$$\begin{aligned} -\int_{s_{1}}^{t_{1}}2uu'V\,\mathrm{d}t\geq{}& \int_{s_{1}}^{t_{1}}u^{2}\rho \Biggl[\sum ^{m}_{i=1}\alpha_{i}Q_{i}+r \sigma^{2}-(r\sigma)' \Biggr]\,\mathrm{d}t \\ &{}+\int _{s_{1}}^{t_{1}} \biggl(\frac{\rho'}{\rho}+2\sigma \biggr)Vu^{2}\,\mathrm{d}t +\int_{s_{1}}^{t_{1}} \frac{V^{2}}{\rho r}u^{2}\,\mathrm{d}t. \end{aligned}$$

That is,

$$\int_{s_{1}}^{t_{1}} \biggl[\frac{u^{2}V^{2}}{\rho r}+2uV \biggl(u'+u \biggl(\frac{\rho'}{2\rho}+\sigma \biggr) \biggr) \biggr] \,\mathrm{d}t +\int_{s_{1}}^{t_{1}}u^{2}\rho \Biggl[ \sum^{m}_{i=1}\alpha_{i}Q_{i}+r \sigma^{2}-(r\sigma )' \Biggr]\,\mathrm{d}t \leq0. $$

Hence,

$$\begin{aligned} &\int_{s_{1}}^{t_{1}} \biggl[\frac{uV}{\sqrt{\rho r}} +\sqrt{ \rho r} \biggl(u'+\frac{u\rho'}{2\rho}+u\sigma \biggr) \biggr]^{2}\,\mathrm{d}t \\ &\quad{} +\int_{s_{1}}^{t_{1}} \Biggl[u^{2}\rho \Biggl(\sum^{m}_{i=1}\alpha_{i}Q_{i}+r \sigma ^{2}-(r\sigma)' \Biggr) -\rho r \biggl(u'+\frac{u\rho'}{2\rho}+u\sigma \biggr)^{2} \Biggr]\,\mathrm{d}t\leq0, \end{aligned}$$

which is equivalent to

$$ \int_{s_{1}}^{t_{1}} \biggl[ \frac{uV}{\sqrt{\rho r}} +\sqrt{\rho r} \biggl(u'+\frac{u\rho'}{2\rho}+u\sigma \biggr) \biggr]^{2}\,\mathrm{d}t+J_{1}(u, \rho, \sigma)\leq0, $$

(2.8)

where $J_{1}(u, \rho, \sigma)$ is as in (2.1). Since $J_{1}(u, \rho, \sigma)>0$, inequality (2.8) yields

$$\int_{s_{1}}^{t_{1}} \biggl[\frac{uV}{\sqrt{\rho r}} +\sqrt{ \rho r} \biggl(u'+\frac{u\rho'}{2\rho}+u\sigma \biggr) \biggr]^{2}\,\mathrm{d}t\leq-J_{1}(u, \rho, \sigma)< 0, $$

which is a contradiction. This contradiction proves that x is not eventually positive.

When x is eventually negative, we use $u\in B_{2}$ and $F(t)\geq0$ on $[s_{2}, t_{2}]$ to arrive at a similar contradiction. The proof is complete. □

Example 2.1

For $t\geq1$, consider the forced second-order neutral delay differential equation

$$ \biggl(x(t)+\frac{1}{2}x \biggl(\frac{t}{2} \biggr) \biggr)''+ 8x (t )+4t^{2}x \biggl( \frac{t}{2} \biggr)=\sin t. $$

(2.9)

Let $r(t)=1$, $P(t)=1/2$, $\tau(t)=t/2$, $m=l=1$, $Q_{1}(t)=8$, $R_{1}(t)=4t^{2}$, $f_{1}(y)=g_{1}(y)=y$, $\alpha_{1}=\beta_{1}=1$, $u=\sin t$, $\rho(t)=1$, and $\sigma(t)=0$. Set $s_{1}=(2n+1)\pi$, $t_{1}=(2n+2)\pi$, $s_{2}=(2n+3)\pi$, and $t_{2}=(2n+4)\pi$. Then

$$\begin{aligned} J_{1}(u, \rho, \sigma) =&\int_{s_{1}}^{t_{1}} \Biggl\{ \rho \Biggl[u^{2} \Biggl(\sum^{m}_{i=1} \alpha _{i}Q_{i}+r\sigma^{2}-(r \sigma)' \Biggr) -r \biggl(u'+\frac{u\rho'}{2\rho}+u\sigma \biggr)^{2} \Biggr] \Biggr\} (t) \,\mathrm{d}t \\ =&\int_{(2n+1)\pi}^{(2n+2)\pi} \bigl(8\sin^{2}t- \cos^{2}t \bigr)\,\mathrm{d}t =\frac{7}{2}\pi. \end{aligned}$$

Similarly, $J_{2}(u, \rho, \sigma)=7\pi/2$. Hence, by Theorem 2.1, every solution of (2.9) is oscillatory.