# Oscillation of certain higher-order neutral partial functional differential equations

- Wei Nian Li
^{1}Email author and - Weihong Sheng
^{1}

**Received: **31 January 2016

**Accepted: **5 April 2016

**Published: **14 April 2016

## Abstract

In this paper, we study the oscillation of certain higher-order neutral partial functional differential equations with the Robin boundary conditions. Some oscillation criteria are established. Two examples are given to illustrate the main results in the end of this paper.

## Keywords

## Mathematics Subject Classification

## Background

It is well known that the theory of partial functional differential equations can be applied to many fields, such as population dynamics, cellular biology, meteorology, viscoelasticity, engineering, control theory, physics and chemistry (Wu 1996). In the monograph, Wu (1996) provided some fundamental theories and applications of partial functional differential equations.

The oscillation theory as a part of the qualitative theory of partial functional differential equations has been developed in the past few years. Many researchers have established some oscillation results for partial functional differential equations. For example, see the monograph (Yoshida 2008) and the papers (Bainov et al. 1996; Fu and Zhuang 1995; Li and Cui 1999; Li 2000; Li and Cui 2001; Ouyang et al. 2005; Gao and Luo 2008; Li and Han 2006; Wang et al. 2010). We especially note that the monograph (Yoshida 2008) contained large material on oscillation theory for partial differential equations.

*n*is an odd integer and \(s\le m\), with the boundary conditions (

*B*1), (

*B*2) and

*M*-space \({\mathbb{R}}^M\), \(\alpha ,\beta \in C(\partial \Omega ,[0,\infty )),\) \(\alpha ^2(x)+\beta ^2(x)\ne 0\), and

*N*is the unite exterior normal vector to \(\partial \Omega\).

- (C1)
\(\mu \in C^{n}([0,\infty );[0,\infty )), 0\le \mu (t)\le 1,\tau =\) const.\({>}0;\)

- (C2)
\(a,a_k\in C([0,\infty );[0,\infty )),\rho _k\in C([0,\infty );[0,\infty )),\rho _k(t)\le t,\) \(\lim _{t\rightarrow +\infty }\rho _k(t)=+\infty , \ k\in I_s=\{1,2,\ldots ,s\};\)

- (C3)
\(p\in C([0,\infty )\times [a,b];[0,\infty )),\) \(g\in C([0,\infty )\times [a,b];[0,\infty )),\) \(g(t,\xi )\le t,\xi \in [a,b],\) \(g(t,\xi )\) is a nondecreasing function with respect to

*t*and \(\xi\), respectively, and \(\lim _{t\rightarrow +\infty }\inf _{\xi \in [a,b]}\{g(t,\xi )\}=+\infty ;\) - (C4)
\(\sigma \in ([a,b];{\mathbb{R}})\) and \(\sigma (\xi )\) is nondecreasing in \(\xi\), the integral in (1) is Stieltjes integral.

As it is customary, the solution \(u(x,t)\in C^n(G)\bigcap C^1(\overline{G})\) of the problem (1), (2) is said to be oscillatory in the domain \(G\equiv \Omega \times [0,\infty )\) if for any positive number \(\mu\) there exists a point \((x_0,t_0)\in \Omega \times [\mu ,\infty )\) such that the equality \(u(x_0,t_0)=0\) holds.

To the best of our knowledge, no result is known regarding the oscillatory behavior of higher-order partial functional differential equations with the Robin boundary condition (2) up to now.

The paper is organized as follows. In “Main results” section, we establish some results for the oscillation of the problem (1), (2). In “Examples” section, we construct two examples to illustrate our main results.

## Main results

In this section, we establish the oscillation criteria of the problem (1), (2). First, we introduce the following lemma which is very useful for establishing our main results.

###
**Lemma 1**

*Suppose that*\(\lambda _0\)

*is the smallest eigenvalue of the problem*

*and*\(\varphi (x)\)

*is the corresponding eigenfunction of*\(\lambda _0\).

*Then*\(\lambda _0=0, \varphi (x)=1\)

*as*\(\beta (x)=0\) \((x\in \Omega )\)

*and*\(\lambda _0>0,\varphi (x)>0\) \((x\in \Omega )\)

*as*\(\beta (x)\not \equiv 0\) \((x\in \partial \Omega ).\)

Next, we give our main results.

###
**Theorem 2**

*If*\(\beta (x)\not \equiv 0\)

*for*\(x\in \partial \Omega\),

*then the necessary and sufficient condition for all solutions of the problem*(1), (2)

*to oscillate is that all solutions of the differential equation*

*to oscillate, where*\(\lambda _0\)

*is the smallest eigenvalue of*(3).

###
*Proof*

(i) Sufficiency. Suppose to the contrary that there is a non-oscillatory solution *u*(*x*, *t*) of the problem (1), (2) which has no zero in \(\Omega \times [t_0,\infty )\) for some \(t_0\ge 0\). Without loss of generality we assume that \(u(x,t)>0, \ u(x,t-\tau )>0, \ u(x,\rho _k(t))>0,\)
\(u(x,g(t,\xi ))>0\), \((x,t)\in \Omega \times [t_1,\infty ),k\in I_s.\)

*x*over the domain \(\Omega\), we have

Obviously, it follows from (9) that *V*(*t*) is a positive solution of Eq. (4), which contradicts the fact that all solutions of Eq. (4) are oscillatory.

Hence \(\widetilde{u}(x,t)=\widetilde{V}(t)\varphi (x)>0\) is a non-oscillatory solution of the problem (1), (2), which is a contradiction. The proof is complete. \(\square\)

###
*Remark 3*

Theorem 2 shows that the oscillation of problem (1), (2) is equivalent to the oscillation of the differential equation (4).

###
**Theorem 4**

*If*\(\beta (x)\not \equiv 0\)

*for*\(x\in \partial \Omega\),

*and the neutral differential inequality*

*has no eventually positive solutions, then every solution of the problem*(1), (2)

*is oscillatory in*

*G*.

###
*Proof*

*u*(

*x*,

*t*) of the problem (1), (2) which has no zero in \(\Omega \times [t_0,\infty )\) for some \(t_0\ge 0\). Without loss of generality we assume that \(u(x,t)>0, \ u(x,t-\tau )>0, \ u(x,\rho _k(t))>0,\) \(u(x,g(t,\xi ))>0\), \((x,t)\in \Omega \times [t_1,\infty ),k\in I_s.\) As in the proof of Theorem 2, we obtain Eq. (9). By Lemma 1, from (9) we have

Using Theorems 1 and 2 in Li and Cui (2001), we can obtain the following two conclusions, respectively.

###
**Theorem 5**

*Assume that*\(\beta (x)\not \equiv 0\)

*for*\(x\in \partial \Omega\).

*If for*\(t_0>0,\)

*then every solution of the problem*(1), (2)

*is oscillatory in*

*G*.

###
**Theorem 6**

*Assume that*\(\beta (x)\not \equiv 0\)

*for*\(x\in \partial \Omega\), \(\mu (t)\equiv \mu\)

*is a positive constant,*\(p(t,\xi )\)

*is periodic in t with period*\(\rho\).

*If for*\(t_0>0,\)

*then every solution of the problem*(1), (2)

*is oscillatory in*

*G*.

## Examples

In this section, we give two examples to illustrate our main results.

###
*Example 7*

###
*Example 8*

## Conclusions

This paper provides some oscillation criteria for solutions of higher-order neutral partial functional differential equations with Robin boundary conditions. Using Lemma 1, we obtain Theorems 2 and 4. We should note that Theorem 2 shows that the oscillation of the problem (1), (2) is equivalent to the oscillation of the functional differential equation (4). Using the results in Li and Cui (2001), two useful conclusions are established in Theorems 5 and 6.

## Declarations

### Authors’ contributions

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

### Acknowledgements

The authors are grateful to anonymous referees for their kind comments and suggestions on this paper. This work is supported by the National Natural Science Foundation of China (10971018).

### Competing interests

Both authors declare that they have no competing interests.

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

## References

- Bainov D, Cui BT, Minchev E (1996) Forced oscillation of solutions of certain hyperbolic equations of neutral type. J Comput Appl Math 72:309–318View ArticleGoogle Scholar
- Fu X, Zhuang W (1995) Oscillation of certain neutral delay parabolic equations. J Math Anal Appl 191:473–489View ArticleGoogle Scholar
- Gao Z, Luo L (2008) Oscillation of solutions of nonlinear neutral hyperbolic partial differential equations with continuous deviating arguments and damped terms. Math Appl 21:399–403 (in Chinese)Google Scholar
- Li WN (2000) Oscillation properties for systems of hyperbolic differential equations of neutral type. J Math Anal Appl 248:369–384View ArticleGoogle Scholar
- Li WN, Cui BT (1999) Oscillation of solutions of neutral partial functional differential equations. J Math Anal Appl 234:123–146View ArticleGoogle Scholar
- Li WN, Cui BT (2001) A class of even order neutral differential inequalities and its applications. Appl Math Comput 122:95–106View ArticleGoogle Scholar
- Li WN, Han M (2006) Oscillation tests for certain systems of parabolic differential equations with neutral type. Rocky Mt J Math 36:1285–1300View ArticleGoogle Scholar
- Ouyang Z, Zhou S, Yin F (2005) Oscillation for a class of odd-order delay parabolic differential equations. J Comput Appl Math 175:305–319View ArticleGoogle Scholar
- Wang CY, Wang S, Yan X, Li L (2010) Oscillation of a class of partial functional population model. J Math Anal Appl 368:32–42View ArticleGoogle Scholar
- Wu J (1996) Theory and applications of partial functional differential equations. Springer, New YorkView ArticleGoogle Scholar
- Ye QX, Li ZY (1990) Theory of reaction diffusion equations. Science Press, Beijing (in Chinese)Google Scholar
- Yoshida N (2008) Oscillation theory of partial differential equations. World Scientific Publishing, SingaporeView ArticleGoogle Scholar