Theory and Modern Applications

# On certain inequalities and their applications in the oscillation theory

## Abstract

In the paper, we offer a set of inequalities involving delayed argument and offer their application for higher-order differential equations of the form

${x}^{\left(n\right)}\left(t\right)+q\left(t\right)x\left(\tau \left(t\right)\right)=0$

to be oscillatory. The conditions obtained essentially improve many other known results.

MSC:34K11, 34C10.

## 1 Introduction

The paper is organized as follows. In the first part we consider only properties of functions and their derivatives, and later we connect the estimate obtained with properties of solutions of differential equations. We shall investigate the properties of a couple of functions $\tau \left(t\right)\in C\left(I\right)$ and $x\left(t\right)\in {C}^{\ell }\left(I\right)$, $I=\left[{t}_{0},\mathrm{\infty }\right)$.

Lemma 1 Assume that is a positive integer such that

eventually. Then for any constant $\lambda \in \left(0,1\right)$ and for every $i=1,2,\dots ,\ell$,

$\frac{{t}^{i}{x}^{\left(i\right)}\left(t\right)}{i!}<\frac{1}{\lambda }\left(\genfrac{}{}{0}{}{\ell }{i}\right)x\left(t\right),$
(1.1)

eventually.

Proof Assume that ${\overline{C}}_{\ell }$ holds for $t\ge {t}_{0}$. Using the monotonicity of ${x}^{\left(\ell \right)}\left(t\right)$, it is easy to see that for any $k\in \left(0,1\right)$,

${x}^{\left(\ell -1\right)}\left(t\right)>{\int }_{{t}_{0}}^{t}{x}^{\left(\ell \right)}\left(s\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}s\ge \left(t-{t}_{0}\right){x}^{\left(\ell \right)}\left(t\right)\ge kt{x}^{\left(\ell \right)}\left(t\right),$
(1.2)

eventually, let us say, for $t\ge {t}_{1}\ge {t}_{0}$. We define a sequence of functions ${\left\{{\rho }_{i}\left(t\right)\right\}}_{1}^{\ell }$ as follows:

$\begin{array}{c}\begin{array}{r}{\rho }_{1}\left(t\right)={x}^{\left(\ell -1\right)}\left(t\right)-kt{x}^{\left(\ell \right)}\left(t\right),\\ {\rho }_{2}\left(t\right)=2{x}^{\left(\ell -2\right)}\left(t\right)-kt{x}^{\left(\ell -1\right)}\left(t\right),\end{array}\hfill \\ ⋮\hfill \\ {\rho }_{\ell }\left(t\right)=\ell x\left(t\right)-kt{x}^{\prime }\left(t\right).\hfill \end{array}$

It follows from (1.2) that ${\rho }_{1}\left(t\right)>0$. An integration of this from ${t}_{1}$ to t yields

${x}^{\left(\ell -2\right)}\left(t\right)\left[1+k\right]-kt{x}^{\left(\ell -1\right)}\left(t\right)\ge c=k\left({x}^{\left(\ell -2\right)}\left({t}_{1}\right)-{t}_{1}{x}^{\left(\ell -1\right)}\left({t}_{1}\right)\right).$

On the other hand, since ${x}^{\left(\ell -2\right)}\left(t\right)\to \mathrm{\infty }$ as $t\to \mathrm{\infty }$, we see that

${x}^{\left(\ell -2\right)}\left(t\right)\left[1-k\right]>c.$

Combining the last two inequalities, we conclude that

${\rho }_{2}\left(t\right)>0.$

Proceeding as above, we verify that ${\rho }_{i}\left(t\right)>0$, eventually, for all $i=1,2,\dots ,\ell -1$. Therefore,

$\begin{array}{c}\ell x\left(t\right)>kt{x}^{\prime }\left(t\right),\hfill \\ \left(\ell -1\right){x}^{\prime }\left(t\right)>kt{x}^{″}\left(t\right),\hfill \\ ⋮\hfill \\ {x}^{\left(\ell -1\right)}\left(t\right)>kt{x}^{\left(\ell \right)}\left(t\right),\hfill \end{array}$

or in other words

$\begin{array}{c}t{x}^{\prime }\left(t\right)<\frac{1}{k}\ell x\left(t\right),\hfill \\ {t}^{2}{x}^{″}\left(t\right)<\frac{1}{{k}^{2}}\ell \left(\ell -1\right)x\left(t\right),\hfill \\ ⋮\hfill \\ {t}^{\ell }{x}^{\left(\ell \right)}\left(t\right)<\frac{1}{{k}^{\ell }}\ell !x\left(t\right).\hfill \end{array}$

Setting $\lambda ={k}^{\ell }$, the last inequalities imply (1.1) and the proof is complete. □

Lemma 2 Assume that $\tau \left(t\right)\le t$ and that is a positive integer such that ${\overline{C}}_{\ell }$ holds. Then, for any constant $\lambda \in \left(0,1\right)$,

$x\left(\tau \left(t\right)\right)\ge \lambda {\left(\frac{\tau \left(t\right)}{t}\right)}^{\ell }x\left(t\right),$
(1.3)

eventually.

Proof Taylor’s theorem implies that

$\begin{array}{rcl}x\left(t\right)& \le & x\left(\tau \left(t\right)\right)+{x}^{\prime }\left(\tau \left(t\right)\right)\left(t-\tau \left(t\right)\right)+\cdots +{x}^{\left(\ell -2\right)}\left(\tau \left(t\right)\right)\frac{{\left(t-\tau \left(t\right)\right)}^{\ell -1}}{\left(\ell -1\right)!}\\ +{x}^{\left(\ell \right)}\left(\tau \left(t\right)\right)\frac{{\left(t-\tau \left(t\right)\right)}^{\ell }}{\ell !}.\end{array}$

Employing (1.1), we have

$\begin{array}{rcl}x\left(t\right)& \le & \frac{x\left(\tau \left(t\right)\right)}{\lambda }\left\{1+\left(\genfrac{}{}{0}{}{\ell }{1}\right)\left(\frac{t}{\tau \left(t\right)}-1\right)+\left(\genfrac{}{}{0}{}{\ell }{2}\right){\left(\frac{t}{\tau \left(t\right)}-1\right)}^{2}+\cdots +\left(\genfrac{}{}{0}{}{\ell }{\ell }\right){\left(\frac{t}{\tau \left(t\right)}-1\right)}^{\ell }\right\}\\ =& \frac{x\left(\tau \left(t\right)\right)}{\lambda }{\left[1+\left(\frac{t}{\tau \left(t\right)}-1\right)\right]}^{\ell }=\frac{1}{\lambda }{\left(\frac{t}{\tau \left(t\right)}\right)}^{\ell }x\left(\tau \left(t\right)\right).\end{array}$

The proof is complete. □

The obtained estimates can be used, e.g., in the theory of functional equations. In the paper, we present their application in discussing oscillatory and asymptotic properties of higher-order delay differential equations.

## 2 Main results

We consider higher-order delay differential equation

where

(H1) $q\left(t\right)>0$, $\tau \left(t\right)\le t$.

Denote by $\mathcal{N}$ the set of all nonoscillatory solutions of (E). It follows from the classical lemma of Kiguradze [1] that the set $\mathcal{N}$ has the following decomposition:

$\begin{array}{c}\mathcal{N}={\mathcal{N}}_{0}\cup {\mathcal{N}}_{2}\cup \cdots \cup {\mathcal{N}}_{n-1},\phantom{\rule{1em}{0ex}}n\text{-odd,}\hfill \\ \mathcal{N}={\mathcal{N}}_{1}\cup {\mathcal{N}}_{3}\cup \cdots \cup {\mathcal{N}}_{n-1},\phantom{\rule{1em}{0ex}}n\text{-even,}\hfill \end{array}$

where the nonoscillatory solution $x\left(t\right)$, let us say positive, satisfies

$x\left(t\right)\in {\mathcal{N}}_{\ell }\phantom{\rule{1em}{0ex}}⇔\phantom{\rule{1em}{0ex}}\left\{\begin{array}{c}{x}^{\left(i\right)}\left(t\right)>0;\phantom{\rule{1em}{0ex}}i=0,1,\dots ,\ell ,\hfill \\ {\left(-1\right)}^{i-\ell }{x}^{\left(i\right)}\left(t\right)>0;\phantom{\rule{1em}{0ex}}i=\ell ,\dots ,n-1.\hfill \end{array}$

A nonoscillatory solution $x\left(t\right)$ of (E) is said to be of degree if $x\left(t\right)\in {\mathcal{N}}_{\ell }$. Following Kondratiev and Kiguradze, we say that (E) has property (A) provided that

$\begin{array}{c}\mathcal{N}={\mathcal{N}}_{0},\phantom{\rule{1em}{0ex}}n\text{-odd,}\hfill \\ \mathcal{N}=\mathrm{\varnothing },\phantom{\rule{1em}{0ex}}n\text{-even.}\hfill \end{array}$

The investigation of oscillatory properties of the second- and higher-order linear differential equations started with the Sturm comparison theorem [2]. Later Mahfoud [3] essentially contributed to the subject and presented a very useful comparison technique for studying the properties of a delay differential equation from those of a differential equation without delay. A new impetus to investigation in this direction was given by papers of Chanturia and Kiguradze [4], Kusano and Naito [5] and Koplatadze et al. [6, 7]. See also [120]. In the paper, we employ Lemma 2 to establish new criteria for oscillation of (E).

It is interesting to note that the condition

${\int }^{\mathrm{\infty }}{\tau }^{n-1}\left(t\right)q\left(t\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}t=\mathrm{\infty }$
(2.1)

is necessary for property (A) of (E). This fact has been observed in [6] and [7].

Theorem 1 Assume that (E) has a solution of degree $\ell >0$, then for any $\lambda \in \left(0,1\right)$ so does the ordinary equation

Proof Assume that (E) possesses a nonoscillatory solution $x\left(t\right)\in {\mathcal{N}}_{\ell }$. We may assume that $x\left(t\right)$ is positive. Then condition (1.3) of Lemma 1 implies that $x\left(t\right)$ is a positive solution of the differential inequality

${x}^{\left(n\right)}\left(t\right)+\lambda {\left(\frac{\tau \left(t\right)}{t}\right)}^{\ell }q\left(t\right)x\left(t\right)\le 0.$

On the other hand, it follows from Theorem 2 of [5] that the corresponding equation (${E}_{\ell }$) has also a solution of degree . The proof is complete. □

So, if we eliminate solutions of degree of equations (${E}_{\ell }$), we get property (A) of studied equation (E). To do it, we recall the following comparison result which is due to Chanturia [4].

Theorem 2 Assume that

${\int }_{t}^{\mathrm{\infty }}q\left(s\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}s\ge {\int }_{t}^{\mathrm{\infty }}p\left(s\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}s.$
(2.2)

If the differential equation

${x}^{\left(n\right)}\left(t\right)+p\left(t\right)x\left(t\right)=0$

has no solution of degree , neither does the equation

${x}^{\left(n\right)}\left(t\right)+q\left(t\right)x\left(t\right)=0.$

In view of Theorem 1, we apply this comparison theorem to equations (${E}_{\ell }$) and the Euler equation

${x}^{\left(n\right)}\left(t\right)+\frac{a}{{t}^{n}}x\left(t\right)=0$
(2.3)

to obtain new criteria for property (A) of (E). Properties of (2.3) are connected with properties of the polynomial ${P}_{n}\left(k\right)=-k\left(k-1\right)\cdots \left(k-n+1\right)$. Let us denote

${M}_{j}=\underset{k\in \left(j-1;j\right)}{max}\left\{{P}_{n}\left(k\right)\right\},$

where $j=2,4,\dots ,n-1$ for n odd, while $j=1,3,\dots ,n-1$ for n even. In other words, ${M}_{j}$ represents all local maxima of the polynomial ${P}_{n}\left(k\right)$ (see Figure 1). Then it is easy to verify (see also [15]) that the following criterion for the ${\mathcal{N}}_{\ell }$ to be empty holds true.

Lemma 3 Let $\ell >0$. If

$a>{M}_{\ell },$
(2.4)

where $\ell =1,3,\dots ,n-1$ for n odd and $\ell =1,3,\dots ,n-1$ for n even, then (2.3) has no solution of degree .

Employing Theorem 2 to (2.3) and (${E}_{\ell }$), in view of Theorem 1, one gets the following theorem.

Theorem 3 Assume that

Then (E) has property (A).

Proof Assume that n is odd. Observing that $\tau \left(t\right)\le t$ and ${M}_{n-1}>{M}_{\ell }$ for every $\ell =2,4,\dots ,n-3$, it follows from (P) that for every $\ell =2,4,\dots ,n-1$,

On the other hand, (${P}_{\ell }$) implies that there exists a couple of constants $\lambda \in \left(0,1\right)$ and $a>{M}_{\ell }$ such that

$\lambda {\int }_{t}^{\mathrm{\infty }}{\left(\frac{\tau \left(s\right)}{s}\right)}^{\ell }q\left(s\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}s>\frac{a}{\left(n-1\right){t}^{n-1}}.$
(2.5)

Since $a>{M}_{\ell }$, Euler equation (2.3) has no solution of degree . On the other hand, taking (2.5) into account, Theorem 2 ensures that (${E}_{\ell }$) has no solution of degree . Finally, Theorem 1 guarantees that (E) has property (A). The proof is complete. □

For $\tau \left(t\right)=\alpha t$, $0<\alpha \le 1$, the previous result simplifies to the following.

Corollary 1 Assume that

Then the delay differential equation

has property (A).

Theorem 4 Let n be odd. Assume that (E) has property (A). Then every nonoscillatory solution $x\left(t\right)$ of (E) satisfies

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

Proof First note that property (A) of (E) implies (2.1). Moreover, it follows from the definition of property (A) that every nonoscillatory solution $x\left(t\right)\in {\mathcal{N}}_{0}$, which implies that there exists ${lim}_{t\to \mathrm{\infty }}x\left(t\right)=c\ge 0$. We claim that $c=0$. If not, then $x\left(\tau \left(t\right)\right)\ge c>0$. An integration of (E) from t to ∞ yields

${x}^{\left(n-1\right)}\left(t\right)\ge {\int }_{t}^{\mathrm{\infty }}q\left(s\right)x\left(\tau \left(s\right)\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}s\ge c{\int }_{t}^{\mathrm{\infty }}q\left(s\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}s.$

Having repeated this procedure, we are led to

$x\left({t}_{1}\right)\ge \frac{c}{\left(n-1\right)!}{\int }_{t}^{\mathrm{\infty }}{\left(s-{t}_{1}\right)}^{n-1}q\left(s\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}s,$

which contradicts (2.1) and we conclude that $c=0$. □

We support our results with the following illustrative example.

Example 1 Consider the fifth-order delay differential equation

The graph of the polynomial $P\left(k\right)=-k\left(k-1\right)\left(k-2\right)\left(k-3\right)\left(k-4\right)$ that corresponds to the fifth-order equation is presented in Figure 1. Employing Matlab, we easily evaluate that

${M}_{4}=3.6314.$

Consequently, criterion (${P}^{\ast }$) for property (A) of (E) reduces for (${E}_{x1}$) to

$a{\alpha }^{4}>3.6314.$

## 3 Comparison

Theorem 2 essentially improves Chanturia’s test [4] that guarantees property (A) of

provided that

$\underset{t\to \mathrm{\infty }}{lim sup}t{\int }_{t}^{\mathrm{\infty }}{s}^{n-2}q\left(s\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}s>\left(n-1\right)!.$

Kiguradze’s test [1] that for property (A) of (E) requires

and Koplatadze’s test [7] for property (A) of (E) that claims the condition

$\begin{array}{c}\underset{t\to \mathrm{\infty }}{lim sup}\left\{\tau \left(t\right){\int }_{\tau \left(t\right)}^{\mathrm{\infty }}{\tau }^{n-2}\left(s\right)q\left(s\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}s+{\int }_{\tau \left(t\right)}^{t}{\tau }^{n-1}\left(s\right)q\left(s\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}s\hfill \\ \phantom{\rule{1em}{0ex}}+\frac{1}{\tau \left(t\right)}{\int }_{{t}_{2}}^{\tau \left(t\right)}s{\tau }^{n-1}\left(s\right)q\left(s\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}s\right\}>\left(n-1\right)!,\hfill \end{array}$
(3.1)

where $\tau \left(t\right)$ is nondecreasing.

We provide details while comparing those criteria with our one.

Example 2 Consider once more the fifth-order delay differential equation (${E}_{x1}$). It is easy to see that Chanturia’s test can be applied only when $\alpha =1$ and requires

$a>4!$

for property (A) of (${E}_{x1}$). Kiguradze’s test fails. On the other hand, Koplatadze’s test simplifies for $\alpha =1$ and $\alpha =0.8$ to

$a>6\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}a>23.9522,$

respectively, while our criterion needs only

$a>3.6314\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}a>8.8658,$

respectively.

## References

1. Kiguradze IT:On the oscillation of solutions of the equation $\frac{{d}^{m}u}{d{t}^{m}}+a\left(t\right){|u|}^{n}signu=0$. Mat. Sb. 1964, 65: 172-187. (Russian)

2. Sturm JCF: Mémoire sur les équations différentielles linéaires du second ordre. J. Math. Pures Appl. 1836, 1: 106-186.

3. Mahfoud WE: Oscillation and asymptotic behavior of solutions of n th order nonlinear delay differential equations. J. Differ. Equ. 1977, 24: 75-98. 10.1016/0022-0396(77)90171-1

4. Kiguradze IT, Chaturia TA: Asymptotic Properties of Solutions of Nonautonomous Ordinary Differential Equations. Kluwer Academic, Dordrecht; 1993.

5. Kusano T, Naito M: Comparison theorems for functional differential equations with deviating arguments. J. Math. Soc. Jpn. 1981, 3: 509-533. Zbl 0494.34049

6. Koplatadze RG: On differential equations with a delayed argument having properties A and B. Differ. Uravn. (Minsk) 1989, 25: 1897-1909.

7. Koplatadze RG, Kvinkadze G, Stavroulakis I: Properties A and B of n th order linear differential equations with deviating argument. Georgian Math. J. 1999, 6: 553-566. 10.1023/A:1022962129926

8. Agarwal RP, Grace SR, O’Regan D: Oscillation Theory for Difference and Functional Differential Equations. Kluwer Academic, Dordrecht; 2000.

9. Baculíková B, Džurina J: Oscillation of third-order neutral differential equations. Math. Comput. Model. 2010, 52: 215-226. 10.1016/j.mcm.2010.02.011

10. Baculíková B, Graef J, Džurina J: On the oscillation of higher order delay differential equations. Nonlinear Oscil. 2012, 15: 13-24.

11. Baculíková B, Džurina J: Oscillation of third-order nonlinear differential equations. Appl. Math. Lett. 2011, 24: 466-470. 10.1016/j.aml.2010.10.043

12. Baculíková B: Properties of third order nonlinear functional differential equations with mixed arguments. Abstr. Appl. Anal. 2011., 2011: Article ID 857860

13. Baculíková B, Džurina J, Rogovchenko Y: Oscillation of third order trinomial differential equations. Appl. Math. Comput. 2012, 218: 7023-7033. 10.1016/j.amc.2011.12.049

14. Baculíková B, Džurina J: Property (A) and oscillation of third order differential equations with mixed arguments. Funkc. Ekvacioj 2012, 55: 239-253.

15. Džurina J: Comparison theorems for nonlinear ODE’s. Math. Slovaca 1992, 42: 299-315.

16. Erbe L, Kong Q, Zhang BG: Oscillation Theory for Functional Differential Equations. Dekker, New York; 1995.

17. Ladde GS, Lakshmikantham V, Zhang BG: Oscillation Theory of Differential Equations with Deviating Arguments. Dekker, New York; 1987.

18. Zafer A: Oscillation criteria for even order neutral differential equations. Appl. Math. Lett. 1998, 11: 21-25.

19. Meng FW, Xu R: Oscillation criteria for certain even order quasi-linear neutral differential equations with deviating arguments. Appl. Math. Comput. 2007, 190: 458-464. 10.1016/j.amc.2007.01.040

20. Li T, Han Z, Zhao P, Sun S: Oscillation of even-order neutral delay differential equations. Adv. Differ. Equ. 2010., 2010: Article ID 184180

## Acknowledgements

This work was supported by the Slovak Research and Development Agency under the contract No. APVV-0008-10.

## Author information

Authors

### Corresponding author

Correspondence to Jozef Džurina.

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

The authors have made the same contribution. All authors read and approved the final manuscript.

## Authors’ original submitted files for images

Below are the links to the authors’ original submitted files for images.

## Rights and permissions

Reprints and permissions

Baculíková, B., Džurina, J. On certain inequalities and their applications in the oscillation theory. Adv Differ Equ 2013, 165 (2013). https://doi.org/10.1186/1687-1847-2013-165