Oscillation of certain higher-order neutral partial functional differential equations

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.


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. Li and Cui (2001) studied the oscillation of even order partial functional differential equations where n ≥ 2 is an even integer, with the two kinds of boundary conditions: and Ouyang et al. (2005) established the oscillation of odd order partial functional differential equations where n is an odd integer and s ≤ m, with the boundary conditions (B1), (B2) and In this paper, we investigate the oscillation of the following higher-order neutral partial functional differential equations with the Robin boundary condition where n ≥ 2 is an even integer, is a bounded domain in R M with a piecewise smooth boundary ∂�, and is the Laplacian in the Euclidean M-space R M , α, β ∈ C(∂�, [0, ∞)), α 2 (x) + β 2 (x) � = 0, and N is the unite exterior normal vector to ∂�.
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 Ye and Li (1990). Suppose that 0 is the smallest eigenvalue of the problem Next, we give our main results.
Theorem 2 If β(x) � ≡ 0 for x ∈ ∂�, 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 0 is the smallest eigenvalue of (3).

From Green's formula and boundary condition (2), it follows that
where dS is the surface element on ∂�.

Combining (5)-(8) 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.
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 β(x) � ≡ 0 for x ∈ ∂�, and the neutral differential inequality has no eventually positive solutions, then every solution of the problem (1), (2) is oscillatory in G.
Using Theorems 1 and 2 in Li and Cui (2001), we can obtain the following two conclusions, respectively.