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# On certain inequalities and their applications in the oscillation theory

- Blanka Baculíková
^{1}and - Jozef Džurina
^{1}Email author

**2013**:165

https://doi.org/10.1186/1687-1847-2013-165

© Baculíková and Džurina; licensee Springer 2013

**Received: **1 March 2013

**Accepted: **23 May 2013

**Published: **11 June 2013

## 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

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

**MSC:**34K11, 34C10.

## Keywords

- higher order differential equations
- delay argument
- oscillation

## 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 $\tau (t)\in C(I)$ and $x(t)\in {C}^{\ell}(I)$, $I=[{t}_{0},\mathrm{\infty})$.

*eventually*.

*Then for any constant*$\lambda \in (0,1)$

*and for every*$i=1,2,\dots ,\ell $,

*eventually*.

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

*t*yields

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

**Lemma 2**

*Assume that*$\tau (t)\le t$

*and that*

*ℓ*

*is a positive integer such that*${\overline{C}}_{\ell}$

*holds*.

*Then*,

*for any constant*$\lambda \in (0,1)$,

*eventually*.

*Proof*Taylor’s theorem implies that

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

where

(H_{1}) $q(t)>0$, $\tau (t)\le t$.

*E*). It follows from the classical lemma of Kiguradze [1] that the set $\mathcal{N}$ has the following decomposition:

*E*) is said to be of degree

*ℓ*if $x(t)\in {\mathcal{N}}_{\ell}$. Following Kondratiev and Kiguradze, we say that (

*E*) has property (A) provided that

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 [1–20]. In the paper, we employ Lemma 2 to establish new criteria for oscillation of (*E*).

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*$\lambda \in (0,1)$

*so does the ordinary equation*

*Proof*Assume that (

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

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

So, if we eliminate solutions of degree *ℓ* 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*

*If the differential equation*

*has no solution of degree*

*ℓ*,

*neither does the equation*

*E*). Properties of (2.3) are connected with properties of the polynomial ${P}_{n}(k)=-k(k-1)\cdots (k-n+1)$. Let us denote

*n*odd, while $j=1,3,\dots ,n-1$ for

*n*even. In other words, ${M}_{j}$ represents all local maxima of the polynomial ${P}_{n}(k)$ (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*

*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* *ℓ*.

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

*Then* (*E*) *has property* (A).

*Proof*Assume that

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

*P*) that for every $\ell =2,4,\dots ,n-1$,

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

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

*has property* (A).

**Theorem 4**

*Let*

*n*

*be odd*.

*Assume that*(

*E*)

*has property*(A).

*Then every nonoscillatory solution*$x(t)$

*of*(

*E*)

*satisfies*

*Proof*First note that property (A) of (

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

*E*) from

*t*to ∞ yields

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

We support our results with the following illustrative example.

*E*) reduces for (${E}_{x1}$) to

## 3 Comparison

*E*) requires

*E*) that claims the condition

where $\tau (t)$ 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

respectively.

## Declarations

### Acknowledgements

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

## Authors’ Affiliations

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