Asymptotic behavior for third-order quasi-linear differential equations

Abstract

In this paper, a class of third-order quasi-linear differential equations withcontinuously distributed delay is studied. Applying the generalized Riccatitransformation, integral averaging technique of Philos type and Young’sinequality, a set of new criteria for oscillation or certain asymptoticbehavior of nonoscillatory solutions of this equations is given. Our resultsessentially improve and complement some earlier publications.

1 Introduction

Consider the following third-order quasi-linear differential equation:

$${[a(t){\left({[x(t)+{\int }_a^{b}p(t,\mu )x[\tau (t,\mu )]d\mu ]}^{″}\right)}^{\gamma }]}^{\prime }+{\int }_c^{d}q(t,\xi )f\left(x[\sigma (t,\xi )]\right)d\xi =0.$$

(1)

We build up the following hypotheses firstly:

(H1) $a(t)\in C([t_0,\mathrm{\infty }),(0,\mathrm{\infty }))$ and ${\int }_{t_0}^{\mathrm{\infty }}a{(s)}^{-\frac{1}{\gamma }}ds=\mathrm{\infty }$;

(H2) $p(t,\mu )\in C([t_0,\mathrm{\infty })\times [a,b],[0,\mathrm{\infty }))$ and $0\le p(t)\equiv {\int }_a^{b}p(t,\mu )d\mu \le p<1$;

(H3) $\tau (t,\mu )\in C([t_0,\mathrm{\infty })\times [a,b],R)$ is not a decreasing function for μ andsuch that

$$\tau (t,\mu )\le t\text{and}\underset{t\to \mathrm{\infty }}{lim}\underset{\mu \in [a,b]}{min}\tau (t,\mu )=\mathrm{\infty };$$

(2)

(H4) $q(t,\xi )\in C([t_0,\mathrm{\infty }),(0,\mathrm{\infty }))$;

(H5) $\sigma (t,\xi )\in C([t_0,\mathrm{\infty })\times [a,b],R)$ is not a decreasing function for ξ andsuch that

$$\sigma (t,\xi )\le t\text{and}\underset{t\to \mathrm{\infty }}{lim}\underset{\xi \in [c,d]}{min}\sigma (t,\xi )=\mathrm{\infty };$$

(3)

(H6) $f(x)\in C(R,R)$ and $\frac{f(x)}{x^{\gamma }}\ge \delta >0$;

(H7) γ is a quotient of odd positive integers.

Define the function by

$$z(t)=x(t)+{\int }_a^{b}p(t,\mu )x[\tau (t,\mu )]d\mu .$$

(4)

A function $x(t)$ is a solution of (1) means that$x(t)\in C^2[T_x,\mathrm{\infty })$, $T_x\ge t_0$, $a(t){(z^{″}(t))}^{\gamma }\in C^1[T_x,\mathrm{\infty })$ and satisfies (1) on $[T_x,\mathrm{\infty })$. In this paper, we restrict our attention to thosesolutions of Eq. (1) which satisfy $sup\{|x(t)|:t\ge T\}>0$ for all $T\ge T_x$. We assume that Eq. (1) possesses such asolution. A solution of Eq. (1) is called oscillatory on $[T_x,\mathrm{\infty })$ if it is eventually positive or eventually negative;otherwise, it is called nonoscillatory.

In recent years, there has been much research activity concerning the oscillationtheory and applications of differential equations; see [1–4] and the reference contained therein. Especially, the study content ofoscillatory criteria of second-order differential equations is very rich. Incontrast, the study of oscillatory criteria of third-order differential equations isrelatively less, but most of works are about delay equations. Some interestingresults have been obtained concerning the asymptotic behavior of solutions of Eq.(1) in the particular case. For example, [5] consider the third-order functional differential equations of the form

$${[a(t){(x^{″}(t))}^{\gamma }]}^{\prime }+q(t)f\left(x[\sigma (t)]\right)=0.$$

(5)

Zhang et al.[6] focus on the following the third-order neutral differential equationswith continuously distributed delay:

$${[a(t){[x(t)+{\int }_a^{b}p(t,\mu )x[\tau (t,\mu )]d\mu ]}^{″}]}^{\prime }+{\int }_c^{d}q(t,\xi )f\left(x[\sigma (t,\xi )]\right)d\xi =0.$$

(6)

Baculíková and Džurina [7] are concerned with the couple of the third-order neutral differentialequations of the form

$${[a(t){\left({[x(t)+p(t)x[\tau (t)]]}^{″}\right)}^{\gamma }]}^{\prime }+q(t)x^{\gamma }[\sigma (t)]=0.$$

(7)

However, as we know, oscillatory behaviors of solutions of Eq. (1) have not beenconsidered up to now. In this paper, we try to discuss the problem of oscillatorycriteria of Philos type of Eq. (1). Applying the generalized Riccati transformation,integral averaging technique of Philos type, Young’s inequality,etc., we obtain some new criteria for oscillation or certain asymptoticbehavior of nonoscillatory solutions of this equations. We should point out thatγ is any quotient of odd positive integers in this paper, but itis required that $\gamma =1$ in [6].

2 Several lemmas

We start our work with the classification of possible nonoscillatory solutions of Eq.(1).

Lemma 2.1 Let$x(t)$be a positive solution of (1), and$z(t)$is defined as in (4). Then$z(t)$has only one of the following two properties eventually:

• (I) $z(t)>0$, $z^{\prime }(t)>0$, $z^{″}(t)>0$;

• (II) $z(t)>0$, $z^{\prime }(t)<0$, $z^{″}(t)>0$.

Proof Let $x(t)$ be a positive solution of (1), eventually (if it iseventually negative, the proof is similar). Then ${[a(t){(z^{″}(t))}^{\gamma }]}^{\prime }<0$. Thus, $a(t){(z^{″}(t))}^{\gamma }$ is decreasing and of one sign and it followshypotheses (H2)-(H7) that there exists $t_1\ge t_0$ such that $z^{″}(t)$ is of fixed sign for $t\ge t_1$. If we admit $z^{″}(t)<0$, then there exists a constant $M>0$ such that

$$z^{″}(t)\le -\frac{M}{a{(t)}^{\frac{1}{\gamma }}},t\ge t_1.$$

(8)

Integrating from $t_1$ to t, we get

$$z^{\prime }(t)\le z^{\prime }(t_1)-M{\int }_{t_1}^{t}a{(s)}^{-\frac{1}{\gamma }}ds.$$

(9)

Let $t\to \mathrm{\infty }$ and using (H1), we have $z^{\prime }(t)\to -\mathrm{\infty }$. Thus $z^{\prime }(t)<0$ eventually, which together with$z^{″}(t)<0$ implies $z(t)<0$, which contradicts our assumption$z(t)>0$. This contradiction shows that$z^{″}(t)>0$, eventually. Therefore $z^{\prime }(t)$ is increasing and thus (I) or (II) holds for$z(t)$, eventually. □

Lemma 2.2 Let$x(t)$be a positive solution of (1), and correspondingly$z(t)$has the property (II). Assume that

$${\int }_{t_0}^{\mathrm{\infty }}{\int }_v^{\mathrm{\infty }}{[\frac{1}{a(u)}{\int }_u^{\mathrm{\infty }}{\int }_c^{d}q(s,\xi )d\xi ds]}^{1/\gamma }dudv=\mathrm{\infty }.$$

(10)

Then

$$\underset{t\to \mathrm{\infty }}{lim}x(t)=0.$$

(11)

Proof Let $x(t)$ be a positive solution of Eq. (1). Since$z(t)$ satisfies the property (II), it is obvious that thereexists a finite limit

$$\underset{t\to \mathrm{\infty }}{lim}z(t)=l.$$

(12)

Next, we claim that $l=0$. Assume that $l>0$, then we have $l<z(t)<l+\epsilon$ for all $\epsilon >0$ and t enough large. Choosing$\epsilon <l(1-p)/p$, we obtain

$$\begin{array}{rl}x(t)& =z(t)-{\int }_a^{b}p(t,\mu )x[\tau (t,\mu )]d\mu \ge l-{\int }_a^{b}p(t,\mu )z[\tau (t,\mu )]d\mu \\ \ge l-p(t)z[\tau (t,a)]\ge l-p(l+\epsilon )\\ =K(l+\epsilon )>Kz(t),\end{array}$$

(13)

where $K=\frac{l-p(l+\epsilon )}{l+\epsilon }>0$. □

Combining (H6), (13) with (1), one can get

$$\begin{array}{rl}{(a(t){[z^{″}(t)]}^{\gamma })}^{\prime }& \le -\delta K^{\gamma }{\int }_c^{d}q(t,\xi ){\left(z[\sigma (t,\xi )]\right)}^{\gamma }d\xi \\ \le -\delta K^{\gamma }{\left(z[\sigma (t,d)]\right)}^{\gamma }{\int }_c^{d}q(t,\xi )d\xi \\ \le -\delta K^{\gamma }{\left(z[{\sigma }_0(t)]\right)}^{\gamma }{q}_1(t),\end{array}$$

(14)

where $q_1(t)={\int }_c^{d}q(t,\xi )d\xi$ and ${\sigma }_0(t)=\sigma (t,d)$. Integrating inequality (14) from t to∞, we get immediately

$$a(t){[z^{″}(t)]}^{\gamma }\ge \delta K^{\gamma }{\int }_t^{\mathrm{\infty }}{q}_1(s){\left(z[{\sigma }_0(s)]\right)}^{\gamma }ds.$$

(15)

Using $z({\sigma }_0(s))>l$, we have

$$\begin{array}{r}{z}^{″}(t)\ge {\delta }^{1/\gamma }Kl{(\frac{1}{a(t)}{\int }_t^{\mathrm{\infty }}{q}_1(s)ds)}^{\frac{1}{\gamma }}\ge {\delta }^{1/\gamma }Kl{(\frac{1}{a(t)}{\int }_t^{\mathrm{\infty }}{\int }_c^{d}q(s,\xi )d\xi ds)}^{\frac{1}{\gamma }};\\ -z^{\prime }(t)\ge {\delta }^{1/\gamma }Kl{\int }_t^{\mathrm{\infty }}{(\frac{1}{a(u)}{\int }_u^{\mathrm{\infty }}{\int }_c^{d}q(s,\xi )d\xi ds)}^{\frac{1}{\gamma }}du;\\ z(t_1)\ge {\delta }^{1/\gamma }Kl{\int }_{t_1}^{\mathrm{\infty }}{\int }_v^{\mathrm{\infty }}{(\frac{1}{a(u)}{\int }_u^{\mathrm{\infty }}{\int }_c^{d}q(s,\xi )d\xi ds)}^{\frac{1}{\gamma }}dudv.\end{array}$$

(16)

We have a contradiction with (10) and so it follows that ${lim}_{t\to \mathrm{\infty }}z(t)=0$, which implies that

$$\underset{t\to \mathrm{\infty }}{lim}x(t)=0.$$

(17)

Lemma 2.3[7]

Assume that$u(t)>0$, $u^{\prime }(t)>0$, $u^{″}(t)<0$on$[t_0,\mathrm{\infty })$. Then, for each$\alpha \in (0,1)$, there exists$T_{\alpha }\ge t_0$such that

$$\frac{u(\sigma (t))}{\sigma (t)}\ge \alpha \frac{u(t)}{t}\mathit{\text{for all }}t\ge T_{\alpha }.$$

(18)

Lemma 2.4[8]

Let$z(t)>0$, $z^{\prime }(t)>0$, $z^{″}(t)>0$, $z^{‴}(t)<0$on$[T_{\alpha },\mathrm{\infty })$. Then there exist$\beta \in (0,1)$and$T_{\beta }\ge T_{\alpha }$such that

$$z(t)\ge \beta t{z}^{\prime }(t)\mathit{\text{for all }}t\ge T_{\beta }.$$

(19)

3 Main results

For simplicity, we introduce the following notations:

$$D=\{(t,s):t\ge s\ge t_0\};D_0=\{(t,s):t>s\ge t_0\}.$$

(20)

A function $H\in C^1(D,R)$ is said to belong to X class$(H\in X)$ if it satisfies

• (i) $H(t,t)=0$, $t\ge t_0$; $H(t,s)>0$, $(t,s)\in D_0$;

• (ii) $\frac{\partial H(t,s)}{\partial s}<0$, there exist $\rho \in C^1([t_0,\mathrm{\infty }),(0,\mathrm{\infty }))$ and $h\in C(D_0,R)$ such that

  $$\frac{\partial H(t,s)}{\partial s}+\frac{{\rho }^{\prime }(t)}{\rho (t)}H(t,s)=-h(t,s){(H(t,s))}^{\frac{\gamma }{1+\gamma }}.$$

  (21)

Theorem 3.1 Assume that (10) holds, there exist$\rho \in C^1([t_0,\mathrm{\infty }),(0,\mathrm{\infty }))$and$H\in X$such that

$$\underset{t\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\frac{1}{H(t,t_0)}{\int }_{t_0}^t[H(t,s)Q(s)-\frac{a(s)\rho (s)h^{\gamma +1}(t,s)}{{(\gamma +1)}^{\gamma +1}}]ds=\mathrm{\infty },$$

(22)

$$Q(s)=\delta {(1-p)}^{\gamma }\rho (s){\left(\frac{\alpha \beta {\sigma }^2(s,c)}{s}\right)}^{\gamma }{\int }_c^{d}q(t,\xi )d\xi .$$

(23)

Suppose, further, that$a^{\prime }(t)>0$. Then every solution$x(t)$of Eq. (1) is either oscillatory or converges to zero.

Proof Assume that Eq. (1) has a nonoscillatory solution$x(t)$. Without loss of generality, we may assume that$x(t)>0$, $t\ge t_1$, $x(\tau (t,\mu ))>0$, $(t,\mu )\in [t_0,\mathrm{\infty })\times [a,b]$, $x(\sigma (t,\xi ))>0$, $(t,\xi )\in [t_0,\mathrm{\infty })\times [c,d]$, $z(t)$ is defined as in (4). By Lemma 2.1, we have that$z(t)$ has the property (I) or the property (II). If$z(t)$ has the property (II). Since (10) holds, then theconditions in Lemma 2.2 are satisfied. Hence ${lim}_{t\to \mathrm{\infty }}x(t)=0$.

When $z(t)$ has the property (I), we obtain

$$\begin{array}{rl}x(t)& =z(t)-{\int }_a^{b}p(t,\mu )x[\tau (t,\mu )]d\mu \ge z(t)-{\int }_a^{b}p(t,\mu )z[\tau (t,\mu )]d\mu \\ \ge z(t)-z[\tau (t,b)]{\int }_a^{b}p(t,\mu )d\mu \ge (1-p)z(t).\end{array}$$

(24)

Using (H5) and (H6), we have

$${(a(t){[z^{″}(t)]}^{\gamma })}^{\prime }\le -\delta {(1-p)}^{\gamma }{\left(z[{\sigma }_1(t)]\right)}^{\gamma }{q}_1(t),$$

(25)

where $q_1(t)={\int }_c^{d}q(t,\xi )d\xi$ and ${\sigma }_1(t)=\sigma (t,c)$. Let

$$w(t)=\rho (t)a(t){\left(\frac{z^{″}(t)}{z^{\prime }(t)}\right)}^{\gamma },t\ge t_1.$$

(26)

Then

$$\begin{array}{r}{w}^{\prime }(t)-\frac{{\rho }^{\prime }(t)}{\rho (t)}w(t)\\ \le -\delta {(1-p)}^{\gamma }{q}_1(t){\left(\frac{z[{\sigma }_1(t)]}{z^{\prime }(t)}\right)}^{\gamma }-\gamma {\left(\frac{1}{a(t)\rho (t)}\right)}^{1/\gamma }{w}^{\frac{\gamma +1}{\gamma }}(t).\end{array}$$

(27)

Choosing $u(t)=z^{\prime }(t)$ in Lemma 2.2, we obtain

$$\frac{1}{z^{\prime }(t)}\ge \frac{\alpha {\sigma }_1(t)}{t{z}^{\prime }({\sigma }_1(t))},t\ge T_{\alpha }\ge t_1.$$

(28)

Using Lemma 2.3, we get

$$z({\sigma }_1(t))\ge \beta {\sigma }_1(t)z^{\prime }({\sigma }_1(t))t\ge T_{\beta }\ge T_{\alpha }.$$

(29)

Combining with (27)-(29), we have

$$w^{\prime }(t)\le -Q(t)+\frac{{\rho }^{\prime }(t)}{\rho (t)}w(t)-\gamma {\left(\frac{1}{a(t)\rho (t)}\right)}^{1/\gamma }{w}^{\frac{\gamma +1}{\gamma }}(t),$$

(30)

where $Q(t)$ is defined by (23). Let

$$A(t)=\frac{{\rho }^{\prime }(t)}{\rho (t)},B(t)=\gamma {\left(\frac{1}{a(t)\rho (t)}\right)}^{1/\gamma }.$$

For $t\ge t_2\ge T_{\beta }$, we have

$$\begin{array}{rl}{\int }_{t_2}^{t}H(t,s)Q(s)ds& \le {\int }_{t_2}^{t}H(t,s)[-w^{\prime }(s)+A(s)w(s)-B(s)w^{\frac{\gamma +1}{\gamma }}(s)]ds\\ =H(t,t_2)w(t_2)-{\int }_{t_2}^t[h(t,s)F(t,s)+B(s){(F(t,s))}^{\frac{\gamma +1}{\gamma }}]ds,\end{array}$$

(31)

where $F(t,s)=w(s)H^{\frac{\gamma }{\gamma +1}}(t,s)$. By Young’s inequality

$$\frac{{(B^{\frac{\gamma }{\gamma +1}}(s)F(t,s))}^{\frac{\gamma +1}{\gamma }}}{\frac{\gamma +1}{\gamma }}+\frac{{(\gamma B^{-\frac{\gamma }{\gamma +1}}(s)\frac{h(t,s)}{\gamma +1})}^{\gamma +1}}{\gamma +1}\ge \frac{\gamma }{\gamma +1}|h(t,s)|F(t,s),$$

(32)

we obtain

$$B(s)F^{\frac{\gamma +1}{\gamma }}(t,s)\ge |h(t,s)|F(t,s)-\frac{a(s)\rho (s)h^{\gamma +1}(t,s)}{{(\gamma +1)}^{\gamma +1}}.$$

(33)

Applying (33) to inequality (31), we obtain

$$\begin{array}{rl}{\int }_{t_2}^{t}H(t,s)Q(s)ds\le & H(t,t_2)w(t_2)+{\int }_{t_2}^t\frac{a(s)\rho (s)h^{\gamma +1}(t,s)}{{(\gamma +1)}^{\gamma +1}}ds\\ -{\int }_{t_2}^t[h(t,s)+|h(t,s)|]F(t,s)ds.\end{array}$$

(34)

Therefore, we have

$$w(t_2)\ge \frac{1}{H(t,t_2)}{\int }_{t_2}^t[H(t,s)Q(s)ds-\frac{a(s)\rho (s)h^{\gamma +1}(t,s)}{{(\gamma +1)}^{\gamma +1}}]ds.$$

(35)

The last inequality contradicts (22). □

Theorem 3.2 Assume that other conditions of Theorem 3.1 are satisfied exceptcondition (22). Further, for every T, the following inequalities hold:

$$0<\underset{s\ge T}{inf}\underset{t\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\frac{H(t,s)}{H(t,T)}\le \mathrm{\infty }$$

(36)

and

$$\underset{t\to \mathrm{\infty }}{lim\hspace{0.17em}sup}{\int }_T^t\frac{a(s)\rho (s)h^{\gamma +1}(t,s)}{H(t,T)}ds<\mathrm{\infty }.$$

(37)

If there exists $\psi \in C([t_0,\mathrm{\infty }),R)$ such that

$$\underset{t\to \mathrm{\infty }}{lim\hspace{0.17em}sup}{\int }_T^t{\left[\frac{{\psi }_{+}^{\gamma +1}(s)}{\rho (s)a(s)}\right]}^{1/\gamma }ds=\mathrm{\infty },$$

(38)

$$\underset{t\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\frac{1}{H(t,T)}{\int }_T^t[H(t,s)Q(s)-\frac{a(s)\rho (s)h^{\gamma +1}(t,s)}{{(\gamma +1)}^{\gamma +1}}]ds\ge \psi (T),$$

(39)

where${\psi }_{+}(s)=max\{\psi (s),0\}$, then every solution$x(t)$of Eq. (1) is either oscillatory or converges to zero.

Proof As the proof of Theorem 3.1, we can see that (31) holds. It followsthat

$$\begin{array}{r}\underset{t\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\frac{1}{H(t,t_2)}{\int }_{t_2}^t(H(t,s)Q(s)-G(t,s))ds\\ \le w(t_2)-\underset{t\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\frac{1}{H(t,t_2)}{\int }_{t_2}^t[h(t,s)F(t,s)+B(s){(F(t,s))}^{\frac{\gamma +1}{\gamma }}+G(t,s)]ds,\end{array}$$

(40)

where $G(t,s)=\frac{a(s)\rho (s)h^{\gamma +1}(t,s)}{{(\gamma +1)}^{\gamma +1}}$.

By (45), we get

$$\psi (t_2)\le w(t_2)-\underset{t\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\frac{1}{H(t,t_2)}{\int }_{t_2}^t[h(t,s)F(t,s)+B(s){(F(t,s))}^{\frac{\gamma +1}{\gamma }}+G(t,s)]ds,$$

(41)

and hence

$$\begin{array}{rl}0& \le \underset{t\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\frac{1}{H(t,t_2)}{\int }_{t_2}^t[h(t,s)F(t,s)+B(s){(F(t,s))}^{\frac{\gamma +1}{\gamma }}+G(t,s)]ds\\ \le w(t_2)-\psi (t_2)<\mathrm{\infty }.\end{array}$$

(42)

Define the functions $\alpha (t)$ and $\beta (t)$ as follows:

$$\begin{array}{r}\alpha (t)=\frac{1}{H(t,t_2)}{\int }_{t_2}^{t}h(t,s)F(t,s)ds,\\ \beta (t)=\frac{1}{H(t,t_2)}{\int }_{t_2}^{t}B(s){(F(t,s))}^{\frac{\gamma +1}{\gamma }}ds.\end{array}$$

(43)

From (37) and (42), we obtain

$$\underset{t\to \mathrm{\infty }}{lim\hspace{0.17em}inf}[\alpha (t)+\beta (t)]<\mathrm{\infty }.$$

(44)

The remainder of the proof is similar to the theorem given in [9–11] and hence is omitted. If $z(t)$ has the property (II), since (10) holds, by Lemma2.2, we have ${lim}_{t\to \mathrm{\infty }}x(t)=0$. □

Theorem 3.3 If we replace (37) by

$$\underset{t\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\frac{1}{H(t,t_0)}{\int }_{t_0}^{t}H(t,s)Q(s)ds<\mathrm{\infty },$$

(45)

and assume that the other assumptions of Theorem 3.2 hold,then every solution of Eq. (1) is either oscillatory or convergesto zero.

Proof The proof is similar to Theorem 3.2 and hence isomitted. □

Remark 3.4 When $\gamma =1$, Theorems 3.1-3.3 with condition (37) reduce toTheorems 3.1-3.3 of Zhang [6], respectively.