On the asymptotic behavior of the solutions of third order delay differential equations

Abstract

By constructing a Lyapunov functional, we obtain some sufficient conditions which guarantee the stability and boundedness of solutions for some nonlinear differential equations of third order with delay. Our result improve and generalize existing results in the relevant literature of nonlinear third order differential equations.

1 Introduction

We consider nonlinear third order delay differential equation of the form

$$\begin{aligned}{}[ \Psi (x)x^{\prime }] ^{\prime \prime }+a(t)x^{\prime \prime }+b(t)\Phi (x)x^{\prime }+c(t)f(x(t-r))=e(t), \end{aligned}$$

(1.1)

where $r>0$, and the functions $a(t), \, b(t),\, c(t), \, e(t),\, f(x), \quad \Psi (x),$ and $\Phi (x)$ are continuous in their respective arguments and $f^{\prime }(x), \quad \Psi ^{\prime }(x), \Phi ^{\prime }(x)$ exist and are continuous for all $x$.

The asymptotic property of solutions of third order differential equations has received a considerable amount of attention. In numerous places in the literature, for example [1–21], the authors dealt with the problems by considering Lyapunov functions or functionals and obtained the criteria for the stability.

In 1974, Hara [8] investigated the asymptotic behavior of solutions of the differential equation without delay of the form

$$\begin{aligned} x'''+a(t)x^{\prime \prime }+b(t)x^{\prime }+c(t)f(x)=e(t), \end{aligned}$$

(1.2)

and showed that all solutions of the Eq. (1.2) are uniformly bounded and satisfy $x(t)\rightarrow 0,\, x'(t)\rightarrow 0$ and $x''(t)\rightarrow 0$. More recently in 2005, Sadek in [13] establishes conditions under which all solutions of third order differential equation with delay of the form,

$$\begin{aligned} x'''+a(t)x^{\prime \prime }+b(t)x^{\prime }+c(t)f(x(t-r))=0 \end{aligned}$$

tend to the zero solution as $t\rightarrow \infty$. Our objective in this paper is to extend the results verified by Sadek [13] to obtain sufficient conditions for the stability and the boundedness of solutions of delay differential equation (1.1) for the cases $e(t)\equiv 0$ and $e(t)\ne 0$. Clearly the equation discussed in Sadek [13] is a special case of Eq. (1.1) when $\Psi (x)=\Phi (x)=1$. We shall use appropriate Lyapounov function and impose suitable conditions on the functions $f(x)$, $\Psi (x)$ and $\Phi (x)$. On the other hand, we can find the same result for the Eq. (1.1) without delay by putting $r=0$, witch is generalization of Hara [8] results.

2 Preliminaries

First, we will give some basic definitions and important stability criteria for the general non-autonomous delay differential system. We consider

$$\begin{aligned} x'=f(t,x_{t}),\quad x_{t}(\theta )=x(t+\theta ), \quad -r\le \theta \le 0 ,\, \, t\ge 0, \end{aligned}$$

(2.1)

where $f:I\times C_{H}\rightarrow \mathbb {R}^{n}$ is a continuous mapping, $f(t,0)=0$, $C_{H}:=\{\phi \in (C[-r,0], \, \mathbb {R}^{n}): \Vert \phi \Vert \le H\}$, and for $H_{1}<H$, there exists $L(H_{1})>0$, with $|f(t,\phi )|<L(H_{1})$ when $\Vert \phi \Vert <H_{1}$.

Definition 2.1

[5] An element $\psi \in C$ is in the $\omega -limit$ set of $\phi$, say $\Omega (\phi )$, if $x(t,0,\phi )$ is defined on $[0,+\infty )$ and there is a sequence $\{t_{n}\}, {t_{n}}\rightarrow \infty$, as $n\rightarrow \infty$, with $\Vert x_{t_{n}}(\phi )-\psi \Vert \rightarrow 0$ as $n\rightarrow \infty$ where $x_{t_{n}}(\phi )=x(t_{n}+\theta ,0,\phi )\, for$ $-r \le \theta \le 0$.

Definition 2.2

[5] A set $Q\subset C_{H}$ is an invariant set if for any $\phi \in Q$, the solution of (2.1), $x(t,0,\phi )$, is defined on $[0,\infty )$ and $x_{t}(\phi )\in Q$ for $t \in [0,\infty )$.

Lemma 2.3

[3] If $\phi \in C_{H}$is such that the solution $x_{t}(\phi )$ of (2,1) with $x_{0}(\phi )=\phi$ is defined on $[0,\infty )$ and $\Vert x_{t}(\phi )\Vert \le H_{1} <H$ for $t\in [0,\infty )$, then $\Omega (\phi )$ is a non-empty, compact, invariant set and

$$\begin{aligned} dist(x_{t}(\phi ),\Omega (\phi ))\rightarrow 0 \quad as \quad t\rightarrow \infty . \end{aligned}$$

Lemma 2.4

[3] let $V(t,\phi ):I\times C_{H}\rightarrow \mathbb {R}$ be a continuous functional satisfying a local Lipschitz condition. $V(t,0)=0$, and such that:

• (i) $W_{1}(|\phi (0)|)\le V(t,\phi ) \le W_{2}(\Vert \phi \Vert )$ where $W_{1}(r)$, $W_{2}(r)$ are wedges.

• (ii) $V'_{(2,1)}(t,\phi )\le 0$, for $\phi \in C_{H}$.

Then the zero solution of (2.1) is uniformly stable.

If $Z=\{\phi \in C_{H}:V'_{(2,1)}(t,\phi )=0\}$, then the zero solution of (2.1) is asymptotically stable, provided that the largest invariant set in Z is $Q=\{0\}$.

3 Assumptions and main results

First, we state some assumptions on the functions that appeared in (1.1). Suppose that there are positive constants $a_{0},\, b_{0},\, c_{0},$ $\psi _{0},\, \psi _{1},\, \phi _{0},\, \phi _{1}, \, A,\, B,\, C,\, N_{1}$, $\delta _{0}$, and $\delta _{1}$ such that the followings conditions are satisfied

• (i) $\displaystyle 0<\, a_{0}\le a(t)\le A;$ $\displaystyle 0<b_{0}\le b(t)\le B;$ $\displaystyle 0< c_{0}\le c(t)\le C$, $t\ge 0,$

• (ii) $\displaystyle 0<\psi _{0}\le \Psi (x)\le \psi _{1}$ and $\displaystyle 0<\phi _{0}\le \Phi (x)\le \phi _{1}$ for all $x,$

• (iii) $\displaystyle f(0)=0,\, \frac{f(x)}{x}\ge \delta _{0}>0$ $(x\ne 0),$ and $\displaystyle |f^{\prime }(x)|\le \delta _{1}$, for all $x,$

• (iv) $\displaystyle \int _{-\infty }^{+\infty }\left| \Psi ^{\prime }(u)\right| du <\infty$ and $\displaystyle \int _{-\infty }^{+\infty }\left| \Phi ^{\prime }(u)\right| du <\infty ,$

• (v) $\displaystyle \int _{0}^{\infty }\left| c^{\prime }(s)\right| ds\le N_{1}< \infty$ and $c^{\prime }(t)\rightarrow 0$ as $t\rightarrow \infty$.

For the case $e(t)\equiv 0$, the following result is introduced.

Theorem 3.1

In addition to conditions (i)–(v) being satisfied, suppose that the following conditions hold

• (H1) $\displaystyle \frac{\psi _{1}C}{b_{0}\phi _{0}}\delta _{1}<\mu <a_{0}$,

• (H2) $\displaystyle \mu a^{\prime }(t)+\Psi (x)\Phi (x)b^{\prime }(t)-\Psi ^{2} (x)\frac{\delta _{1}}{\mu }c^{\prime }(t)<\mu b_{0}\phi _{0}-\psi _{1} C\delta _{1}$.

Then every solution of (1.1) is uniformly asymptotically stable, provided that

$$\begin{aligned} r<\min \left\{ \frac{2(a_{0}-\mu )}{\psi _{1}C\delta _{1}},\frac{\psi _{0}^{3}(\mu b_{0}\phi _{0}-\psi _{1}C\delta _{1})}{\psi _{1}^{2}C\delta _{1}(\mu +\mu \psi _{0}^{2}+\psi _{0})}\right\} . \end{aligned}$$

Proof

We use the following differential system which is equivalent to Eq. (1.1)

$$\begin{aligned}&x^{\prime }=\frac{1}{\Psi (x)}y,\nonumber \\&y^{\prime }=z, \\&z^{\prime }=-\frac{a(t)}{\Psi (x)}z+\frac{a(t)\Psi ^{\prime }(x)}{\Psi ^{3}(x)}y^{2} -\frac{b(t)\Phi (x)y}{\Psi (x)}-c(t)f(x) \nonumber \\&\quad \quad \quad +\int _{t-r}^{t}y(s)\frac{f^{\prime }(x(s))}{\Psi (x(s))}\nonumber . \end{aligned}$$

(3.1)

The proof depend on some fundamental properties of a continuously differentiable Lyapunov functional $V=V(t,x,y,z)$ defined as

$$\begin{aligned} V(t,x_{t},y_{t},z_{t})=&\mu c(t)F(x)+c(t)f(x)y+\frac{1}{2}\frac{b(t)\Phi (x)}{\Psi (x)}y^{2} +\frac{\mu a(t)}{2\Psi ^{2}(x)}y^{2}\\&+\frac{\mu }{\Psi (x)}yz+\frac{1}{2}z^{2}+\lambda \int _{-r}^{0}\int _{t+s}^{t}y^{2} (\xi )d\xi ds, \end{aligned}$$

such that $F(x)=\int _{0}^{x}f(u)du$, and $\lambda$ is a positive constant which will be determined later in the proof. To show that V is a positive function, we rewrite V above thus

$$\begin{aligned} V(t,x_{t},y_{t},z_{t})=\mu c(t)G(x,y)+V_{1}+V_{2}+\lambda \int _{-r}^{0}\int _{t+s}^{t}y^{2} (\xi )d\xi ds, \end{aligned}$$

(3.2)

where

$$\begin{aligned} G(x,y)&= F(x)+\frac{1}{\mu }yf(x)+\frac{\delta _{1}}{2\mu ^{2} }y^{2},\\ V_{1}&= V_{1}(t,x_{t},y_{t},z_{t})=\frac{1}{2}\left[ -\frac{c(t)\delta _{1}}{\mu }+\frac{b(t)\Phi (x)}{\Psi (x)}\right] y^{2},\\ V_{2}&= V_{2}(t,x_{t},y_{t},z_{t})=\frac{\mu a(t)}{2\Psi ^{2} (x)}y^{2}+\frac{\mu }{\Psi (x)}yz+\frac{1}{2}z^{2}. \end{aligned}$$

By using hypotheses, we obtain

$$\begin{aligned} \mu c(t)G(x,y)&= \mu c(t)\left[ F(x)+\frac{\delta _{1}}{2\mu ^{2} }\left( y+\frac{\mu }{\delta _{1}}f(x)\right) ^{2} -\frac{1}{2\delta _{1} }f^{2}(x) \right] \\&\ge \mu c(t)\left[ \int _{0}^{x} \left( 1-\frac{f^{\prime }(u)}{\delta _{1} }\right) f(u)du\right] \ge 0. \end{aligned}$$

$V_{2}$ can be rearranged as the following

$$\begin{aligned} V_{2}(t,x_{t},y_{t},z_{t})&= \frac{1}{2}\frac{\mu a(t)}{\Psi ^{2}(x)}y^{2} +\frac{\mu }{\Psi (x)}yz+\frac{1}{2}z^{2}\\&= \frac{1}{2}\left( z+\frac{\mu }{\Psi (x)}y\right) ^{2}-\frac{1}{2}\frac{\mu ^{2}}{\Psi ^{2}(x)}y^{2}+\frac{1}{2}\frac{\mu a(t)}{\Psi ^{2}(x)}, \end{aligned}$$

from hypothesis $(H_{1})$, $\displaystyle a_{0}-\mu >0,$ then $\displaystyle \frac{a(t)\mu }{\Psi ^{2}(x)}-\frac{\mu ^{2}}{\Psi ^{2}(x) }>0$, it follows that there is a positive constant $k_{1}$ such that

$$\begin{aligned} V_{2}(t,x_{t},y_{t},z_{t})\ge k_{1}(y^{2} +z^{2} ), \end{aligned}$$

from which we deduce that $V_{2}$ is positive definite. Furthermore, from hypotheses (i) and (ii), we obtain

$$\begin{aligned} V_{1}(t,x_{t},y_{t},z_{t})\ge \frac{1}{2}\left[ \frac{b_{0}\phi _{0}\mu -\psi _{1}C\delta _{1}}{\mu \psi _{1}}\right] y^{2}. \end{aligned}$$

Hence, it is evident from (H1) and the terms contained in the last inequality, that there exist sufficiently small positive constant $k_{2}$, such that

$$\begin{aligned} V_{1}+V_{2}\ge k_{2}(y^{2} +z^{2} ). \end{aligned}$$

Using (3.2) we get

$$\begin{aligned} V\ge \mu c_{0}G(x,y)+k_{2}(y^{2} +z^{2}). \end{aligned}$$

(3.3)

Therefore we can find a continuous function $W_{1}(|\varphi (0)|)$ with

$$\begin{aligned} W_{1}(|\varphi (0)|)\ge 0 \quad \text {and} \quad W_{1}(|\varphi (0)|)\le V(t,\varphi ). \end{aligned}$$

The existence of a continuous function $W_{2}(\Vert \varphi \Vert )$ which satisfies the inequality $V(t,\varphi )\le W_{2}(\Vert \varphi \Vert )$, is easily verified.

The derivative of the Lyapunov functional $V(t,x_{t},\!y_{t},\!z_{t})$, along a solution $(x(t),\,\! y(t),\! \, z(t))$ of the system (3.1), with respect to $t$ is after simplifying

$$\begin{aligned}&V'_{(3.1)}=\mu c^{\prime }(t)F(x)+c^{\prime }(t)yf(x)+\frac{c^{\prime }(t)\delta _{1}}{2\mu }y^{2}+(a(t)-\mu )\alpha (t)zy\\&\qquad \qquad +\frac{b(t)}{2}\beta (t)y^{2}+\left( \frac{c(t)f^{\prime }(x)}{\Psi (x)}-\frac{\mu b(t)\Phi (x)}{\Psi ^{2}(x)}\right) y^{2}\\&\qquad \qquad +\left( \frac{1}{2}\frac{\mu a^{\prime }(t)}{\Psi ^{2}(x)} +\frac{b^{\prime }(t)\Phi (x)}{2\Psi (x)} -\frac{c^{\prime }(t)\delta _{1}}{2\mu }\right) y^{2}+\left( \frac{\mu -a(t)}{\Psi (x)}\right) z^{2}+\lambda r y^{2} \\&\qquad \qquad +c(t)\left( z+\frac{\mu }{\Psi (x)}y\right) \int _{t-r}^{t}y(s)\frac{f^{\prime }(x(s))}{\Psi (x(s))}ds-\lambda \int _{t-r}^{t} y^{2}(\xi )d\xi , \end{aligned}$$

where

$$\begin{aligned} \alpha (t)=\frac{\Psi '(x(t))}{\Psi ^{2}(x(t))}x'(t), \quad \beta (t)=\frac{\Psi (x)\Phi ^{\prime }(x)-\Phi (x)\Psi ^{\prime }(x)}{\Psi ^{2}(x) }x^{\prime }(t). \end{aligned}$$

By the assumptions (i)–(iii), (H1)–(H2), and using the Schwartz inequality $2|uv|\le u^{2} +v^{2}$ we find

$$\begin{aligned}&V'_{(3.1)}\le \mu c^{\prime }(t)\left[ F(x)+\frac{1}{\mu }yf(x)+\frac{\delta _{1}}{ 2\mu ^{2} }y^{2} \right] \\&\qquad \qquad +\frac{1}{\psi _{1}}(\mu -a_{0})z^{2}+\left[ \frac{\psi _{1}C\delta _{1}-\mu b_{0}\phi _{0}}{\psi _{1}^{2}}+\lambda r\right] y^{2}\\&\qquad \qquad +\frac{1}{2}\left( (A-\mu )\left| \alpha (t)\right| +B\left| \beta (t)\right| \right) (y^{2} +z^{2} )\\&\qquad \qquad +\frac{1}{2\psi _{1}^{2}}\left[ \mu a^{\prime }(t)+b^{\prime }(t)\Phi (x)\Psi (x)- \Psi ^{2}(x) \frac{c^{\prime }(t)\delta _{1}}{\mu }\right] y^{2} \\&\qquad \qquad +c(t)\left( z+\frac{\mu }{\Psi (x)}y\right) \int _{t-r}^{t}y(s)\frac{f^{\prime }(x(s))}{\Psi (x(s))}ds-\lambda \int _{t-r}^{t} y^{2}(\xi )d\xi . \end{aligned}$$

Taking $k_{3}=\frac{1}{2} \max \{A-\mu ,B\}$ then

$$\begin{aligned}&V'_{(3.1)}\le \mu c^{\prime }(t)G(x,y)+\left[ \frac{\psi _{1}C\delta _{1}-\mu b_{0}\phi _{0}}{\Psi ^{2}(x)}+\lambda r\right] y^{2}\\&\qquad \qquad +\frac{1}{2\psi _{1}^{2} }\left[ \mu a^{\prime }(t)+b^{\prime }(t)\Phi (x)\Psi (x)- \Psi ^{2}(x) \frac{\delta _{1}}{\mu }c^{\prime }(t)\right] y^{2}\\&\qquad \qquad +\frac{1}{\psi _{1}}(\mu -a_{0})z^{2}+k_{3}(\left| \alpha (t)\right| +\left| \beta (t)\right| ) (y^{2} +z^{2})\\&\qquad \qquad +c(t)\left( z+\frac{\mu }{\Psi (x)}y\right) \int _{t-r}^{t}y(s)\frac{f^{\prime }(x(s))}{\Psi (x(s))}ds-\lambda \int _{t-r}^{t} y^{2}(\xi )d\xi . \end{aligned}$$

From (iii) $\vert f^{\prime }(x)\vert \le \delta _{1},$ and using the Schwartz inequality again we have

$$\begin{aligned} \frac{\mu c(t)}{\Psi (x)}y\int _{t-r}^{t}\frac{y(s)}{\Psi (x)}f^{\prime }(x(s))ds\le \frac{C\delta _{1}\mu r}{2\psi _{0}}y^{2} +\frac{C\mu \delta _{1}}{2\psi _{0}^{3}}\int _{t-r}^{t}y^{2} (\xi )d\xi , \end{aligned}$$

and

$$\begin{aligned} c(t)z\int _{t-r}^{t}\frac{y(s)}{\Psi (x)}f^{\prime }(x(s))ds\le \frac{C\delta _{1}r}{2}z^{2} +\frac{C\delta _{1}}{2\psi _{0}^{2} }\int _{t-r}^{t}y^{2} (\xi )d\xi , \end{aligned}$$

from which we deduce that

$$\begin{aligned}&V'_{(3.1)}\le \mu c^{\prime }(t)G(x,y)+\frac{1}{2\psi _{1}^{2} }\left[ \mu a^{\prime }(t)+b^{\prime }(t)\Phi (x)\Psi (x)- \Psi ^{2}(x) \frac{\delta _{1}}{\mu }c^{\prime }(t)\right] y^{2}\\&\qquad \qquad \qquad +\left[ \frac{\psi _{1}C\delta _{1}-\mu b_{0}\phi _{0}}{\psi _{1}^{2}}+\lambda r+\frac{C\delta _{1}\mu r}{2\psi _{0}}\right] y^{2}+\left[ \frac{1}{\psi _{1}}(\mu -a_{0})+\frac{C\delta _{1}r}{2}\right] z^{2}\\&\qquad \qquad \qquad +k_{3}(\vert \alpha (t) \vert + \vert \beta (t)\vert ) (y^{2} +z^{2}) + \left[ \frac{C\delta _{1}}{2\psi _{0}^{2} }\left( 1+\frac{\mu }{\psi _{0}}\right) -\lambda \right] \int _{t-r}^{t}y^{2} (\xi )d\xi . \end{aligned}$$

Choosing $\displaystyle \frac{C\delta _{1}}{2\psi _{0}^{2} }\left( 1+\frac{\mu }{\psi _{0}}\right) =\lambda$, and using condition (H1) we get

$$\begin{aligned}&V'_{(3.1)} \le \mu c^{\prime }(t)G(x,y)- \left[ \frac{\mu b_{0}\phi _{0}-\psi _{1}C\delta _{1}}{2\psi _{1}^{2} } -\frac{C\delta _{1}}{2\psi _{0} }\left( \mu +\frac{1}{\psi _{1}}+\frac{\mu }{\psi _{0}^{2}}\right) r\right] y^{2} \\&\qquad \qquad \qquad -\left[ \frac{a_{0}-\mu }{\psi _{1}}-\frac{C\delta _{1}r}{2}\right] z^{2}+ k_{3}(\vert \alpha (t)\vert +|\beta (t)|) (y^{2} +z^{2}). \end{aligned}$$

We define the Lyapounov functional $W=W(t,x_{t},y_{t},z_{t})$ as

$$\begin{aligned} W(t,x_{t},y_{t},z_{t})=(\exp -\eta (t))V(t,x_{t},y_{t},z_{t})=(\exp -\eta (t))V, \end{aligned}$$

where

$$\begin{aligned} \eta (t)=\int _{0}^{t}\left[ \frac{1}{\gamma }(\vert \alpha (s)\vert +\vert \beta (s)\vert )+\frac{1}{c_{0}}|c'(s)|\right] ds, \end{aligned}$$

and $\gamma$ is a positive constant which will be determined later in the proof. It is easily verified that

$$\begin{aligned} W'_{(3.1)}(t,x_{t},y_{t},z_{t})=(\exp -\eta (t)) \left[ V'_{(3.1)}-\left( \frac{1}{\gamma }(\vert \alpha (t)\vert +\vert \beta (t)\vert )+\frac{1}{c_{0}}|c'(t)|\right) V\right] , \end{aligned}$$

from conditions (ii) and (iv) we obtain

$$\begin{aligned} \int _{0}^{t} \vert \alpha (s)ds\vert&= \int _{0}^{t}\left| \frac{\Psi ^{\prime }(x(s))}{\Psi ^{2}(x(s))}x^{\prime }(s)\right| ds\\&= \int _{\omega _{1}(t)}^{\omega _{2}(t)}\left| \frac{\Psi ^{\prime }(u)}{\Psi ^{2}(u)}\right| du\le \frac{1}{\psi _{0}^{2}}\int _{\omega _{1}(t)}^{\omega _{2}(t)}\vert \Psi ^{\prime }(u)\vert du\\&< \frac{1}{\psi _{0}^{2}}\int _{-\infty }^{+\infty }\vert \Psi ^{\prime }(u)\vert du \le N_{2}<\infty , \end{aligned}$$

where $\omega _{1}(t) = \min \{x(0),x(t)\}$, $\omega _{2}(t)=\max \{x(0),x(t)\}$. We get also

$$\begin{aligned} \int _{0}^{t} \vert \beta (s)ds\vert&\le \int _{0}^{t}\left| \Phi (x(s))\frac{\Psi ^{\prime }(x(s))}{\Psi ^{2}(x(s))}x^{\prime }(s)\right| ds +\int _{0}^{t}\left| \frac{ \Phi '(x(s))x^{\prime }(s)}{\Psi (x(s))}\right| ds\\&= \int _{\omega _{1}(t)}^{\omega _{2}(t)}\left| \Phi (u)\frac{\Psi ^{\prime }(u)}{\Psi ^{2}(u)} \right| du + \int _{\omega _{1}(t)}^{\omega _{2}(t)}\left| \frac{ \Phi '(u)}{\Psi (u)}\right| du\\&\le \frac{\phi _{1}}{\psi _{0}^{2}}\int _{\omega _{1}(t)}^{\omega _{2}(t)} \vert \Psi ^{\prime }(u) \vert du + \frac{1}{\psi _{0}} \int _{\omega _{1}(t)}^{\omega _{2}(t)} \vert \Phi '(u)\vert du\\&< \frac{\phi _{1}}{\psi _{0}^{2}}\int _{-\infty }^{+\infty }\vert \Psi ^{\prime }(u)\vert du + \frac{1}{\psi _{0}}\int _{-\infty }^{+\infty }\vert \Phi ^{\prime }(u)\vert du \le N_{3} < \infty . \end{aligned}$$

Using the inequality (3.3) we have

$$\begin{aligned}&V'_{(3.1)}-\left( \frac{1}{\gamma }(\left| \alpha (t)\right| +\left| \beta (t)\right| )+\frac{1}{c_{0}}|c'(t)|\right) V\\&\quad \le -\left[ \frac{\mu b_{0}\phi _{0}-\psi _{1}C\delta _{1}}{2\psi _{1}^{2} } -\frac{C\delta _{1}}{2\psi _{0} }\left( \mu +\frac{1}{\psi _{0}}+\frac{\mu }{\psi _{0}^{2}}\right) r\right] y^{2}-\left[ \frac{a_{0}-\mu }{\psi _{1}}-\frac{C\delta _{1}r}{2}\right] z^{2}\\&\qquad +\left[ \left( k_{3}\left| \alpha (t)\right| -\frac{k_{2}}{\gamma } \left| \alpha (t)\right| \right) +\left( k_{3}\left| \beta (t)\right| -\frac{k_{2}}{\gamma }\left| \beta (t)\right| \right) \right] (y^{2} +z^{2}). \end{aligned}$$

Putting $\displaystyle \gamma =\frac{k_{2}}{k_{3}}$ we obtain

$$\begin{aligned} W'_{(3.1)}\le \!-\!K \left( \left[ \frac{\mu b_{0}\phi _{0}\!-\!\psi _{1}C\delta _{1}}{2\psi _{1}^{2} } \!-\!\frac{C\delta _{1}}{2\psi _{0} }\left( \mu \!+\!\frac{1}{\psi _{0}}\!+\!\frac{\mu }{\psi _{0}^{2}}\right) r\right] y^{2}\!-\!\left[ \frac{a_{0}\!-\!\mu }{\psi _{1}}-\frac{C\delta _{1}r}{2}\right] z^{2}\right) \end{aligned}$$

where $\displaystyle K= \exp -\displaystyle \left( \frac{k_{3}(N_{2}+N_{3})}{k_{2}}+\frac{N_{1}}{c_{0}}\right)$. If we take

$$\begin{aligned} r<\min \left\{ \frac{2(a_{0}-\mu )}{\psi _{1}C\delta _{1}},\frac{\psi _{0}^{3}(\mu b_{0}\phi _{0}-\psi _{1}C\delta _{1})}{\psi _{1}^{2}C\delta _{1}(\mu +\mu \psi _{0}^{2}+\psi _{0})}\right\} \end{aligned}$$

then

$$\begin{aligned} W'_{(3.1)}(t,x_{t},y_{t},z_{t})\le -L(y^{2} +z^{2}) , \quad \text {for some} \quad L>0. \end{aligned}$$

It can also be followed that the largest invariant set in $Z$ is $Q = \{0\}$, where

$$\begin{aligned} Z=\{\phi \in C_{H}: W'_{(3.1)}(\phi )=0 \}. \end{aligned}$$

That is, the only solution of system (3.1) for which $W'_{(3.1)}(t,x_{t},y_{t},z_{t})=0$ is the solution $x = y= z= 0$. The above discussion guarantees that the null solution of Eq. (1.1) is uniformly asymptotically stable.

The proof of the theorem is now completed. $\square$

Example

We consider the following third order delay differential equation

$$\begin{aligned}&\left[ \left( \frac{\cos (x)}{1+x^{2}}+4\right) x^{\prime }(t)\right] ^{\prime \prime }+(\cos t +15)x''(t) \nonumber \\&\quad +\left( \frac{5}{2}-\frac{1}{2}e^{-2t}\right) \left( \frac{\sin (x)}{1+x^{2}}+11\right) x^{\prime }(t)\nonumber \\&\quad +\left( \sin \frac{t}{2} +3\right) \left[ x(t-r)+\frac{x(t-r)}{1+x^{2}(t-r)}\right] =0. \end{aligned}$$

(3.4)

It can be seen that

$$\begin{aligned}&14=a_{0}\le a(t)=\cos t+15\le 16, \quad -1\le a'(t)=-\sin t\le 1, \, t\ge 0,\\&2=b_{0}\le b(t)=\frac{5}{2}-\frac{1}{2}e^{-2t}\le \frac{5}{2},\, 0\le b'(t)=e^{-2t}\le 1,\, t\ge 0,\\&2\le c(t)=\sin \frac{t}{2}+3\le 4=C,\, -\frac{1}{2}\le c'(t)= \frac{1}{2}\cos \frac{t}{2}\le \frac{1}{2}, \, t\ge 0,\\&1\le \frac{f(x)}{x}=1+\frac{1}{1+x^{2}} \, \text {with} \, x\ne 0, \Vert f^{\prime }(x)|\le \delta _{1}=2\, \text {and} \, \mu =8,\\&3 \le \Psi (x)=\frac{\cos (x)}{1+x^{2}}+4\le 5,\\&10=\phi _{0} \le \Phi (x)=\frac{\sin (x)}{1+x^{2}}+11 \le 12. \end{aligned}$$

An easy computations show that conditions (H1) and (H2) are satisfied. Indeed,

$$\begin{aligned} \frac{\psi _{1}C}{b_{0}\phi _{0}}\delta _{1}=2<\mu <a_{0}=14. \end{aligned}$$

We have also

$$\begin{aligned} \mu a'(t)+\Psi (x)\Phi (x) b'(t)-\Psi ^{2}(x)\frac{\delta _{1}}{\mu }c'(t)&\le \mu +60+\frac{25}{\mu }=71.12\\&< \mu b_{0}\phi _{0}-\psi _{1}C\delta _{1}=120. \end{aligned}$$

It is straightforward to verify that

$$\begin{aligned} \int _{-\infty }^{+\infty }\left| \Psi ^{\prime }(u)\right| du&\le \int _{-\infty }^{+\infty }\left[ \left| \frac{\sin u}{1+u^{2}}\right| +\left| \frac{2u\cos u}{(1+u^{2})^{2}}\right| \right] du\\&\le \pi +2. \end{aligned}$$

Similarly,

$$\begin{aligned} \int _{-\infty }^{+\infty }\left| \Phi ^{\prime }(u)\right| du&\le \int _{-\infty }^{+\infty }\left[ \left| \frac{\cos u}{1+u^{2}}\right| +\left| \frac{2u\sin u}{(1+u^{2})^{2}}\right| \right] du\\&\le \pi +2. \end{aligned}$$

Thus all the assumptions of Theorem 3.1. hold, this shows that every solution of (3.4) is uniformly asymptotically stable.

In the case $e(t)\ne 0$ we have the following result:

Theorem 3.2

If the assumptions of Theorem 3.1 hold true, and in addition

$$\begin{aligned} \int _{0}^{t}e(s)ds\le e_{0}<\infty \text { for all }t\ge 0, \end{aligned}$$

then all solutions of the Eq. (1.1) are bounded.

Proof

The remaining of this proof follows the strategy indicated in the proof of Theorem 2 in [12] and hence it omitted. $\square$