Oscillation of solutions to fourth-order delay differential equations with middle term

1. Introduction

In this work, we study asymptotic properties of solutions of differential equation of the fourth order and with middle term

We assume \(r\left( \ell \right) \in C\left( \left[ \ell _{0},\infty \right) ,\mathbb{R}\right) ,\) \(r\left( \ell \right) >0\ \)for all \(\ell \geq \ell _{0},\ p,\) \( q,\) \(\sigma \in C\left( \left[ \ell _{0},\infty \right) ,\mathbb{R}\right) \) such that \(p\left( \ell \right) \geq 0,\ q>0\),\(\ \sigma \left( \ell \right) \leq \ell \) and \(\underset{\ell \rightarrow \infty }{\lim } \sigma \left( \ell \right) =\infty .\) We say that a function \(y\left( \ell \right) \)is a solution of (1), we mean a non-trivial real function \(y(\ell ) \in C\left( [\ell _{y},\infty )\right) \), \(\ell _{y}\geq \ell _{0}\), satisfying (1) on \([\ell _{y},\infty )\) and moreover, having the properties: \(y(\ell )\), \( y^{\prime }(\ell )\), \(y^{\prime \prime }(\ell )\) and \(r\left( \ell \right) \left[ y^{\prime \prime \prime }\left( \ell \right) \right] \) are continuously differentiable for all \(\ell \in \lbrack \ell _{y},\infty )\). We consider only those solutions \(y\left( \ell \right) \) of (1) which satisfy \(\sup \{\left\vert y(\ell )\right\vert :\ell \geq \ell \}>0,\) for any \(\ell \geq \ell _{y}\). A solution of (1) is called oscillatory if it has arbitrary large zeros, otherwise it is called nonoscillatory.

The oscillations of higher and fourth order differential equations have been studied by several authors and several techniques have been proposed for obtaining oscillatory criteria for higher and fourth order differential equations. For treatments on this subject, we refer the reader to the texts [1, 2, 3, 4, 5, 6] and the articles [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]. In what follows, we review some results that have provided the background and the motivation, for the present work.

Hou and Cheng [22] studied the oscillation of differential equation with middle term \begin{align*} y^{\left( 4\right) }\left( \ell \right) +p\left( \ell \right) y^{\prime }\left( \ell \right) +q\left( \ell \right) f\left( y\left( \sigma \left( \ell \right) \right) \right) =0, \ell \geq \ell _{0} \end{align*} under the condition: third-order differential equation \(z^{\prime \prime \prime }\left( \ell \right) +p\left( \ell \right) z\left( \ell \right) =0,\) is nonoscillatory.

Džurina and Jadlovská [11] have considered the differential equation of the fourth order with a negative term \begin{align*} y^{\left( 4\right) }\left( \ell \right) +p\left( \ell \right) y^{\prime }\left( \ell \right) +q\left( \ell \right) y\left( \sigma \left( \ell \right) \right) =0,\ell \geq \ell _{0} \end{align*} under the condition: all solutions of the contributory third-order differential equation are nonoscillatory.

Moaaz et al. [19] studied the oscillatory behavior of the third-order nonlinear differential equation with middle term of the form \begin{align*} \left[ r_{2}\left( \ell \right) \left( r_{1}\left( \ell \right) \left( y^{\prime }\left( \ell \right) \right) ^{\alpha }\right) ^{\prime }\right] ^{\prime }+\phi \left( \ell ,y^{\prime }\left( \sigma \left( \ell \right) \right) \right) +q\left( \ell \right) f\left( y\left( \sigma \left( \ell \right) \right) \right) =0,\ell \geq \ell _{0} \end{align*} under Our aim, in the present work, is to use the generalized Riccati transformations and new comparison principles to obtain a new conditions for the oscillation of every solutions of (1) under condition An example is included to illustrate the main results.

2. Main results

In the next section, we shall establish some oscillation criteria for equation (1). The proof of our main results are essentially based on the following lemmas.

Lemma 1. (See [20], Lemma 1) Let \(\beta \geq 1\ \)be\(\ \)a ratio of two numbers\(,\) where \(U\) and \(V\) are constants. Then \begin{equation*} Uz-Vz^{\frac{\beta +1}{\beta }}\leq \frac{\beta ^{\beta }}{(\beta +1)^{\beta +1}}\frac{U^{\beta +1}}{V^{\beta }},\ V>0. \end{equation*}

Lemma 2. (See [6], Lemma 13) If the function \(u\) satisfies \(u^{(\kappa )}>0,\) \(\kappa =0,1,...,m,\) and \( u^{\left( m+1\right) }< 0,\) then \begin{equation*} \frac{u\left( \ell \right) }{\ell ^{m}/m!}\geq \frac{u^{\prime }\left( \ell \right) }{\ell ^{m-1}/\left( m-1\right) !}. \end{equation*}

Lemma (See [1], Lemma 2.2.3) Let \(h\in C^{n}\left( \left[ \ell _{0},\infty \right) ,\left( 0,\infty \right) \right) .\) Let \( h^{\left( n\right) }\left( \ell \right) \) is of a fixed sign and on the interval \(\left[ \ell _{0},\infty \right) \), \(h^{\left( n\right) }\left( \ell \right) \) not identically zero, and that there exists a \(\ell _{1}\geq \ell _{0}\) such that \(h^{\left( n-1\right) }\left( \ell \right) h^{\left( n\right) }\left( \ell \right) \leq 0\) for all \(\ell \geq \ell _{1}\). If \( \underset{\ell \rightarrow \infty }{\lim }h\left( \ell \right) \neq 0\) then for every \(\lambda \in \left( 0,1\right) \) there exists \(\ell _{\lambda }\geq \ell _{0}\) such that \begin{equation*} h\left( \ell \right) \geq \frac{\lambda }{\left( n-1\right) !}\ell ^{n-1}\left\vert h^{\left( n-1\right) }\left( \ell \right) \right\vert \text{ for }\ell \geq \ell _{\lambda }. \end{equation*}

For convenience, we denote \begin{equation*} \rho _{+}^{\prime }\left( \ell \right) :=\max \left\{ 0,\ \rho _{+}^{\prime }\left( \ell \right) \right\} \text{ and\ }\delta _{+}^{\prime }\left( \ell \right) :=\max \left\{ 0,\ \delta ^{\prime }\left( \ell \right) \right\} . \end{equation*}

Theorem 4. Assume that (4) holds . If there exist positive functions \(\rho ,\delta \in C^{1}\left( \left[ \ell _{0},\infty \right) \right) \ \)such that

for some \(\mu \in \left( 0,1\right) ,\ \)and then all solutions of (1) are oscillatory.

Proof. Assume that (1) has a solution \(y\) is nonoscillatory. Without loss of generality, we may assume that there exists a \(\ell _{1}\in \left[ \ell _{0},\infty \right) \) such that \(y(\ell )>0\), \(y(\sigma (\ell ))>0\) for all \( \ell \in \left[ \ell _{0},\infty \right) \). It\(\ \)follows from (1) and (4) that \(\left( r\left( \ell \right) y^{\prime \prime \prime }\left( \ell \right) \right) ^{\prime }< 0\) and there exist two possible cases for\(\ \ell \geq \ell _{1},\ \)where\(\ \ell _{1}\geq \ell _{0}\ \)is sufficiently large: \begin{eqnarray*} &&\left. \left( C_{1}\right) \ \ \ y^{\left( \kappa \right) }\left( \ell \right) >0 \;\; \text{for} \;\;\kappa =0,1,2,3;\right. \\ &&\left. \left( C_{2}\right) \;\; y^{\left( \kappa \right) }\left( \ell \right) >0\;\; \text{for}\;\; \kappa =0,1,3,\text{ and }y^{\prime \prime }\left( \ell \right) < 0.\right. \end{eqnarray*} Assume that we have Case \(\left( C_{1}\right) \). We define a generalized Riccati substitution by

Then \(\omega \left( \ell \right) >0\). Differentiating, we obtain \begin{eqnarray*} \omega ^{\prime }\left( \ell \right) &=&\rho \left( \ell \right) \left( \frac{r\left( \ell \right) \left( y^{\prime \prime \prime }\left( \ell \right) \right) }{y^{\prime \prime }}\right) ^{\prime }+\rho ^{\prime }\left( \ell \right) \frac{r\left( \ell \right) \left( y^{\prime \prime \prime }\left( \ell \right) \right) }{y^{\prime \prime }\left( \ell \right) } , \\ &=&\rho \left( \ell \right) \frac{\left( r\left( \ell \right) \left( y^{\prime \prime \prime }\left( \ell \right) \right) \right) ^{\prime }}{ y^{\prime \prime }}-\rho \left( \ell \right) \frac{r\left( \ell \right) \left( y^{\prime \prime \prime }\left( \ell \right) \right) ^{2}}{\left( y^{\prime \prime }\right) ^{2}}+\rho ^{\prime }\left( \ell \right) \frac{ r\left( \ell \right) \left( y^{\prime \prime \prime }\left( \ell \right) \right) }{y^{\prime \prime }}. \end{eqnarray*} By virtue of (1), we have. From (7),we see that \begin{equation*} \omega ^{\prime }\left( \ell \right) \leq \rho _{+}^{\prime }\left( \ell \right) \frac{r\left( \ell \right) \left( y^{\prime \prime \prime }\left( \ell \right) \right) }{y^{\prime \prime }}-\rho \left( \ell \right) \frac{ p\left( \ell \right) y^{\prime \prime \prime }\left( \ell \right) }{ y^{\prime \prime }}-\frac{\rho \left( \ell \right) \left( q\left( \ell \right) y\left( \sigma \left( \ell \right) \right) \right) }{y^{\prime \prime }}-\rho \left( \ell \right) \frac{r\left( \ell \right) \left( y^{\prime \prime \prime }\left( \ell \right) \right) ^{2}}{\left( y^{\prime \prime }\right) ^{2}}. \end{equation*} Hence, by (7), we obtain From Lemma 2, we have that \(y\left( \ell \right) \geq \frac{\ell }{2 }y^{\prime }\left( \ell \right) \) and hence, \begin{equation*} \frac{y^{\prime }\left( \ell \right) }{y\left( \ell \right) }\leq \frac{2}{ \ell }. \end{equation*} Integrating from \(\ell \) to \(\sigma \left( \ell \right) \) we find \begin{equation*} \frac{y\left( \sigma \left( \ell \right) \right) }{y\left( \ell \right) } \geq \frac{\sigma ^{2}\left( \ell \right) }{\ell ^{2}}. \end{equation*} This implies that It follows from Lemma 3 that for all \(\mu \in \left( 0,1\right) \) and every sufficiently large \(\ell \). Thus, by (8), (9) and (10), we get \begin{eqnarray*} \omega ^{\prime }\left( \ell \right) &\leq &\frac{\rho _{+}^{\prime }\left( \ell \right) }{\rho \left( \ell \right) }\omega \left( \ell \right) -\frac{ p\left( \ell \right) }{r\left( \ell \right) }\omega \left( \ell \right) -\rho \left( \ell \right) q\left( \ell \right) \frac{\sigma ^{2}\left( \ell \right) }{\ell ^{2}}\frac{y\left( \ell \right) }{y^{\prime \prime }}-\frac{1 }{\rho \left( \ell \right) r\left( \ell \right) }\omega \left( \ell \right) ^{2}, \\ &\leq &\frac{\rho _{+}^{\prime }\left( \ell \right) }{\rho \left( \ell \right) }\omega \left( \ell \right) -\frac{p\left( \ell \right) }{r\left( \ell \right) }\omega \left( \ell \right) -\rho \left( \ell \right) q\left( \ell \right) \left( \frac{\sigma \left( \ell \right) }{\ell }\right) ^{2} \frac{\mu }{2}\ell ^{2}-\frac{1}{\rho \left( \ell \right) r\left( \ell \right) }\omega \left( \ell \right) ^{2}. \end{eqnarray*} This implies that Using Lemma 1 with \(U=\left[ \frac{\rho _{+}^{\prime }\left( \ell \right) }{\rho \left( \ell \right) }-\frac{p\left( \ell \right) }{r\left( \ell \right) }\right] ,V=\frac{1}{\rho \left( \ell \right) r\left( \ell \right) }\) and \(z=\omega \), we get \begin{equation*} \left[ \frac{\rho _{+}^{\prime }\left( \ell \right) }{\rho \left( \ell \right) }-\frac{p\left( \ell \right) }{r\left( \ell \right) }\right] \omega \left( \ell \right) -\frac{1}{\rho \left( \ell \right) r\left( \ell \right) } \omega \left( \ell \right) ^{2}\leq \frac{1}{4\rho \left( \ell \right) r\left( \ell \right) }\left[ \frac{\rho _{+}^{\prime }\left( \ell \right) }{ \rho \left( \ell \right) }-\frac{p\left( \ell \right) }{r\left( \ell \right) }\right] ^{2}. \end{equation*} Hence, we obtain \begin{equation*} \omega ^{\prime }\left( \ell \right) \leq -\rho \left( \ell \right) q\left( \ell \right) \frac{\mu }{2}\sigma ^{2}\left( \ell \right) +\frac{1}{4\rho \left( \ell \right) r\left( \ell \right) }\left[ \frac{\rho _{+}^{\prime }\left( \ell \right) }{\rho \left( \ell \right) }-\frac{p\left( \ell \right) }{r\left( \ell \right) }\right] ^{2}. \end{equation*} Which implies that \begin{equation*} \int_{\ell _{1}}^{\ell }\left( \rho \left( s\right) q\left( s\right) \frac{ \mu }{2}\sigma ^{2}\left( s\right) -\frac{1}{4\rho \left( s\right) r\left( s\right) }\left[ \frac{\rho _{+}^{\prime }\left( s\right) }{\rho \left( s\right) }-\frac{p\left( s\right) }{r\left( s\right) }\right] ^{2}\right) ds\leq \omega \left( \ell _{1}\right) , \end{equation*} for some \(\mu \in \left( 0,1\right) \ \)which contradicts (4). Assume that we have Case \(\left( C_{2}\right) \) holds. Define Then \(\psi \left( \ell \right) >0\ \)for\(\ \ell \geq \ell _{1}\ \)and Integrating (1) from \(\ell \) to \(u\) we find \begin{equation*} r\left( u\right) \left( y^{\prime \prime \prime }\right) \left( u\right) -r\left( \ell \right) \left( y^{\prime \prime \prime }\right) \left( \ell \right) +\int_{\ell }^{u}q\left( s\right) y\left( \sigma \left( s\right) \right) ds\leq 0. \end{equation*} By virtue of\(\ y^{\prime }\left( \ell \right) >0, y(\ell )>0 \)and\(\ y^{\prime \prime }\left( \ell \right) < 0\)by (9), we obtain \begin{equation*} r\left( u\right) \left( y^{\prime \prime \prime }\right) \left( u\right) -r\left( \ell \right) \left( y^{\prime \prime \prime }\right) \left( \ell \right) +\int_{\ell }^{u}q\left( s\right) \left( \frac{\sigma ^{2}\left( s\right) }{s^{2}}\right) y\left( \ell \right) ds\leq 0, \end{equation*} whence it follows, by \(y^{\prime }\left( \ell \right) >0,\) that \begin{equation*} r\left( u\right) \left( y^{\prime \prime \prime }\right) \left( u\right) -r\left( \ell \right) \left( y^{\prime \prime \prime }\right) \left( \ell \right) +y\left( \ell \right) \int_{\ell }^{u}q\left( s\right) \left( \frac{ \sigma ^{2}\left( s\right) }{s^{2}}\right) ds\leq 0. \end{equation*} Letting \(u\rightarrow \infty \), we arrive at the inequality \begin{equation*} -r\left( \ell \right) \left( y^{\prime \prime \prime }\right) \left( \ell \right) +y\left( \ell \right) \int_{\ell }^{\infty }q\left( s\right) \left( \frac{\sigma ^{2}\left( s\right) }{s^{2}}\right) ds\leq 0, \end{equation*} we get we see Hence, by (16) in (14), we find Thus, we have Integrating from \(\ell _{1}\) \ to \(\ell \), we get \begin{equation*} \left. \int_{\ell _{1}}^{\ell }\left[ \delta \left( s\right) \int_{s}^{\infty }\left[ \frac{1}{r\left( \upsilon \right) }\int_{\upsilon }^{\infty }q\left( \nu \right) \left( \tfrac{\sigma ^{2}\left( \nu \right) }{ \nu ^{2}}\right) d\nu \right] d\upsilon -\tfrac{\left( \delta ^{\prime }\left( s\right) \right) ^{2}}{4\delta \left( s\right) }\right] ds\leq \psi \left( \ell _{1}\right) ,\right. \end{equation*} which contradicts (5).
The proof of the theorem is complete.

Another criteria for oscillation of (1) can be establish by comparison with ordinary equations of the lower order. We extend a comparison theorem that fasten properties of solutions of (1) with those of second-order differential equations. It is well known (see [1]Agarwal}) that the differential equation where \(a\) ,\(\ q\in C[\ell _{0}\)\quotesinglbase \(\ \infty ),\ a\left( \ell \right) ,\ q\left( \ell \right) >0\), and the necessary and sufficient condition for nonoscillatory of this equation is to there exist a number \( \ell \geq \ell _{0}\),\ and a function \(\upsilon \) \(\in C^{1}[\ell \) \quotesinglbase\ \(\infty ),\) satisfying \begin{equation*} \upsilon ^{\prime }\left( \ell \right) +a^{-1}\left( \ell \right) \upsilon ^{2}\left( \ell \right) +q\left( \ell \right) \leq 0\text{\quotesinglbase\ \ \ \ on\ }[\ell \text{\quotesinglbase\ }\infty ). \end{equation*} In the following, we compare the behavior of oscillatory of (1) with the half-linear differential equations of type (19).

Lemma 5. \label{L1-y} (see[1]) Let \begin{equation*} \int_{\ell _{0}}^{\infty }\frac{1}{a\left( s\right) }ds=\infty . \end{equation*} Then the condition \begin{equation*} \underset{\ell \rightarrow \infty }{\lim \inf }\left( \int_{\ell _{0}}^{\infty }\frac{1}{a\left( s\right) }ds\right) \int_{\ell }^{\infty }q\left( s\right) ds>\frac{1}{4}, \end{equation*} guarantees oscillation of (19).

Theorem 6. Let (4) and assume that the equation

and are oscillatory, then every solution of (1) is oscillatory.

Proof. Proceeding as in proof of the Theorem 4. If we set \(\rho \left( \ell \right) =1\) in (11), then we get \begin{equation*} \omega ^{\prime }\left( \ell \right) +\frac{p\left( \ell \right) }{r\left( \ell \right) }\omega \left( \ell \right) +q\left( \ell \right) \frac{\mu }{2} \sigma ^{2}\left( \ell \right) +\frac{1}{r\left( \ell \right) }\omega \left( \ell \right) ^{2}\leq 0, \end{equation*} for every constant \(\mu \in \left( 0,\ 1\right) . \)Thus, we can see that equation (20) is nonoscillatory for every constant \(\mu \in \left( 0,\ 1\right) ,\ \)which is a contradiction. If we now set \(\delta \left( \ell \right) =1\) in (17), then we find \begin{equation*} \psi ^{\prime }\left( \ell \right) +\int_{\ell }^{\infty }\left[ \frac{1}{ r\left( \upsilon \right) }\int_{\upsilon }^{\infty }q\left( s\right) \left( \frac{\sigma ^{2}\left( s\right) }{s^{2}}\right) ds\right] d\upsilon +\psi ^{2}\left( \ell \right) \leq 0. \end{equation*} Hence, equation (21) is nonoscillatory, which is a contradiction. Theorem 6 is proved.

3. Example

In this section, we give the following example to illustrate our main results.

Example 1. Consider the differential equation

where \(\beta >0\) is a constant. Let \begin{equation*} r\left( \ell \right) =\frac{1}{\ell },\ p\left( \ell \right) =1\backslash 2\ell ^{2},\ \sigma \left( \ell \right) =\frac{\ell }{2},\ p\left( \ell \right) =1\backslash 2\ell ^{2},\ q\left( \ell \right) =\frac{\beta }{\ell }, \end{equation*} we get \begin{equation*} R\left( \ell \right) =2\backslash 3\left( \ell ^{3\backslash 2}-1\right) \rightarrow \infty \ \text{as }\ell \rightarrow \infty . \end{equation*} If we now set \(\rho \left( s\right) =\delta \left( s\right) =1\) then \begin{equation*} \int_{\ell _{0}}^{\infty }\left( \rho \left( s\right) q\left( s\right) \frac{ \mu }{2}\sigma ^{2}\left( s\right) -\frac{1}{4\rho \left( s\right) r\left( s\right) }\left[ \frac{\rho _{+}^{\prime }\left( s\right) }{\rho \left( s\right) }-\frac{p\left( s\right) }{r\left( s\right) }\right] ^{2}\right) ds, \end{equation*} \begin{equation*} =\left( \frac{\beta \mu }{8}-\frac{1}{4}\right) \int_{\ell _{0}}^{\infty }\ell d\ell =\infty ,\ \text{if\ }\beta >\frac{2}{\mu }\ \text{for some constant }\mu \in \left( 0,\ 1\right) \end{equation*} and \begin{equation*} \int_{\ell _{0}}^{\infty }\left[ \delta \left( s\right) \int_{s}^{\infty } \left[ \frac{1}{r\left( \upsilon \right) }\int_{\upsilon }^{\infty }q\left( \nu \right) \left( \dfrac{\sigma ^{2}\left( \nu \right) }{\nu ^{2}}\right) d\nu \right] d\upsilon -\dfrac{\left( \delta ^{\prime }\left( s\right) \right) ^{2}}{4\delta \left( s\right) }\right] ds=\infty . \end{equation*} Thus, by Theorem 4, every solution of equation (22) is oscillatory, provided \(\beta >\frac{2}{\mu }\).

Conclusion

In this work, by using the generalized Riccati transformations technique and new comparison principles, we offer some new sufficient conditions which ensure that any solution of Equation (1) oscillates under the condition ( 4).\(\ \)Further, we can try to get some oscillation criteria of Equation ( 1) if \(y\left( \ell \right) =x\left( \ell \right) +p\left( \ell \right) x\left( \tau \left( \ell \right) \right) \) in the future work.

Acknowledgments

The authors thank the reviewers for for their useful comments, which led to the improvement of the content of the paper.

Author Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Competing Interests

The author(s) do not have any competing interests in the manuscript.