# Oscillation of fourth-order neutral differential equations with *p*-Laplacian like operators

- Tongxing Li
^{1}Email author, - Blanka Baculíková
^{2}, - Jozef Džurina
^{2}and - Chenghui Zhang
^{3}

**2014**:56

https://doi.org/10.1186/1687-2770-2014-56

© Li et al.; licensee Springer. 2014

**Received: **3 January 2014

**Accepted: **11 February 2014

**Published: **14 March 2014

## Abstract

We study oscillatory behavior of a class of fourth-order neutral differential equations with a *p*-Laplacian like operator using the Riccati transformation and integral averaging technique. A Kamenev-type oscillation criterion is presented assuming that the noncanonical case is satisfied. This new theorem complements and improves a number of results reported in the literature. An illustrative example is provided.

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

## Keywords

*p*-Laplace differential equationnoncanonical operator

## 1 Introduction

*p*-Laplacian like operator

It is interesting to study equation (1.1) since the *p*-Laplace differential equations have applications in continuum mechanics as seen from [1]. Throughout, we assume that $p>1$ is a constant, $\mathbb{I}:=[{t}_{0},\mathrm{\infty})$, $r\in {\mathrm{C}}^{1}(\mathbb{I},(0,\mathrm{\infty}))$, ${r}^{\prime}(t)\ge 0$, $a,\sigma ,{q}_{i},{\tau}_{i}\in \mathrm{C}(\mathbb{I},\mathbb{R})$, $0\le a(t)<1$, ${q}_{i}(t)\ge 0$, $i=1,2,\dots ,l$, $\sigma (t)\le t$, ${lim}_{t\to \mathrm{\infty}}\sigma (t)=\mathrm{\infty}$, there exists a function $\tau \in {\mathrm{C}}^{1}(\mathbb{I},\mathbb{R})$ such that $\tau (t)\le {\tau}_{i}(t)$ for $i=1,2,\dots ,l$, $\tau (t)\le t$, ${\tau}^{\prime}(t)>0$, and ${lim}_{t\to \mathrm{\infty}}\tau (t)=\mathrm{\infty}$.

We use the notation ${t}_{-1}:={min}_{t\in [{t}_{0},\mathrm{\infty})}\{\sigma (t),{\tau}_{1}(t),{\tau}_{2}(t),\dots ,{\tau}_{l}(t)\}$. By a solution of (1.1), we mean a function $x\in \mathrm{C}([{t}_{-1},\mathrm{\infty}),\mathbb{R})$ which has the property $r{|{z}^{\u2034}|}^{p-2}{z}^{\u2034}\in {\mathrm{C}}^{1}(\mathbb{I},\mathbb{R})$ and satisfies (1.1) on $\mathbb{I}$. We consider only those solutions *x* of (1.1) which satisfy $sup\{|x(t)|:t\ge {t}_{\ast}\}>0$ for all ${t}_{\ast}\ge {t}_{0}$ and tacitly assume that (1.1) possesses such solutions. A solution *x* of (1.1) is called oscillatory if it has arbitrarily large zeros on $\mathbb{I}$; otherwise, it is said to be nonoscillatory. Equation (1.1) is termed oscillatory if all its solutions oscillate.

has been studied by Li *et al.* [11]. Note that [[11], Theorem 2.2] ensures that every solution *x* of the studied equation is either oscillatory or tends to zero as $t\to \mathrm{\infty}$ and, unfortunately, cannot distinguish solutions with different behaviors.

It should be noted that research in this paper is strongly motivated by the recent paper [11]. The purpose of this paper is to establish a Kamenev-type theorem which guarantees that all solutions of equation (1.1) are oscillatory in the case where (1.3) holds and without requiring conditions (1.4). In the sequel, all functional inequalities are assumed to hold for all *t* large enough.

## 2 Main results

We begin with the following lemma.

**Lemma 2.1** (See [14])

*Let*$f\in {\mathrm{C}}^{n}(\mathbb{I},{\mathbb{R}}^{+})$.

*Assume that*${f}^{(n)}$

*is eventually of one sign for all large*

*t*,

*and there exists a*${t}_{1}\ge {t}_{0}$

*such that*${f}^{(n)}(t){f}^{(n-1)}(t)\le 0$

*for all*$t\ge {t}_{1}$.

*Then*,

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

*there exist a*${t}_{\lambda}\in [{t}_{1},\mathrm{\infty})$

*and a constant*$M>0$

*such that*

*for all* $t\in [{t}_{\lambda},\mathrm{\infty})$.

**Lemma 2.2** (See [[4], Lemma 2.2.3])

*Let*

*f*

*be as in Lemma*2.1.

*If*${lim}_{t\to \mathrm{\infty}}f(t)\ne 0$,

*then*,

*for every constant*$k\in (0,1)$,

*there exists a*${t}_{k}\in [{t}_{1},\mathrm{\infty})$

*such that*

*for all* $t\in [{t}_{k},\mathrm{\infty})$.

**Theorem 2.3**

*Assume*(1.3)

*and let one of the following conditions hold*:

*and*

*Suppose also that there exist functions*$\rho \in {\mathrm{C}}^{1}(\mathbb{I},(0,\mathrm{\infty}))$, $H,\varrho \in \mathrm{C}(\mathbb{D},\mathbb{R})$,

*where*$\mathbb{D}=\{(t,s):t\ge s\ge {t}_{0}\}$

*such that*

*and*

*H*

*has a nonpositive continuous partial derivative*$\partial H/\partial s$

*satisfying*,

*for all sufficiently large*$T\ge {t}_{0}$,

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

*and for all constants*$M>0$,

*where*

*and*

*If there exist functions*$\delta \in {\mathrm{C}}^{1}(\mathbb{I},(0,\mathrm{\infty}))$, $K,\xi \in \mathrm{C}(\mathbb{D},\mathbb{R})$

*such that*

*and*

*K*

*has a nonpositive continuous partial derivative*$\partial K/\partial s$

*satisfying*,

*for all sufficiently large*$T\ge {t}_{0}$

*and for some constant*$k\in (0,1)$,

*where*

*and*

*then equation* (1.1) *is oscillatory*.

*Proof*Let

*x*be a nonoscillatory solution of (1.1). Without loss of generality, we may assume that

*x*is eventually positive. Equation (1.1) implies that there exists a ${t}_{1}\ge {t}_{0}$ such that the following three possible cases hold for all $t\ge {t}_{1}$:

- (1)
$z(t)>0$, ${z}^{\prime}(t)<0$, ${z}^{\u2033}(t)>0$, ${z}^{\u2034}(t)<0$, ${(r{|{z}^{\u2034}|}^{p-2}{z}^{\u2034})}^{\prime}(t)\le 0$;

- (2)
$z(t)>0$, ${z}^{\prime}(t)>0$, ${z}^{\u2034}(t)>0$, ${z}^{(4)}(t)\le 0$, ${(r{|{z}^{\u2034}|}^{p-2}{z}^{\u2034})}^{\prime}(t)\le 0$;

- (3)
$z(t)>0$, ${z}^{\prime}(t)>0$, ${z}^{\u2033}(t)>0$, ${z}^{\u2034}(t)<0$, ${(r{|{z}^{\u2034}|}^{p-2}{z}^{\u2034})}^{\prime}(t)\le 0$.

We consider each of these cases separately.

*t*to

*ι*, $\iota \ge t\ge {t}_{1}$, we obtain

*t*, we have

*t*to ∞, we get

*t*, we have

which contradicts (2.2).

*t*,

*t*, we obtain

*t*,

which contradicts (2.3).

*t*, we obtain

*t*,

which contradicts (2.4). This completes the proof. □

**Remark 2.4** Choosing different combinations of functions *H*, *ρ*, *K*, and *δ*, one can derive from Theorem 2.3 a variety of efficient tests for oscillation of equation (1.1) and its particular cases.

## 3 Example and discussion

The following example illustrates applications of Theorem 2.3.

**Example 3.1**For $t\ge 1$ and $0\le {a}_{0}<1$, consider the fourth-order neutral differential equation

Let $p=2$, $\tau (t)=t-3\pi $, $\rho (t)=\delta (t)=1$, and $H(t,s)=K(t,s)={(t-s)}^{2}$. It is not difficult to verify that all assumptions of Theorem 2.3 are satisfied, and hence equation (3.1) is oscillatory. As a matter of fact, one such solution is $x(t)=sint$.

**Remark 3.2** Oscillation theorem established in this paper for equation (1.1) complements, on one hand, results reported by Baculíková and Džurina [6], Karpuz [7], and Li *et al.* [9] because we use assumption (1.3) rather than (1.2) and, on the other hand, those by Li *et al.* [10] and Zhang *et al.* [16, 17, 19–21] since our theorem can be applied to the case where $a(t)\ne 0$.

**Remark 3.3** We point out that, contrary to [[11], Theorem 2.2], Theorem 2.3 does not need restrictive conditions (1.4) and can ensure that all solutions of equation (1.1) oscillate, which, in a certain sense, is a significant improvement compared to [[11], Theorem 2.2] for fourth-order neutral differential equations.

**Remark 3.4**It would be of interest to study equation (1.1) in the case where

for future research.

## Declarations

### Acknowledgements

The authors express their sincere gratitude to the anonymous referees for the careful reading of the original manuscript and useful comments that helped to improve the presentation of the results and accentuate important details. This research is supported by the National Key Basic Research Program of P.R. China (2013CB035604) and NNSF of P.R. China (Grant Nos. 61034007, 51277116, 51107069).

## Authors’ Affiliations

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