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Oscillation of certain higherorder neutral partial functional differential equations
SpringerPlus volume 5, Article number: 459 (2016)
Abstract
In this paper, we study the oscillation of certain higherorder 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 \(n\ge 2\) 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 \(s\le m\), with the boundary conditions (B1), (B2) and
In this paper, we investigate the oscillation of the following higherorder neutral partial functional differential equations
with the Robin boundary condition
where \(n\ge 2\) is an even integer, \(\Omega\) is a bounded domain in \({\mathbb{R}}^{M}\) with a piecewise smooth boundary \(\partial \Omega\), and \(\Delta\) is the Laplacian in the Euclidean Mspace \({\mathbb{R}}^M\), \(\alpha ,\beta \in C(\partial \Omega ,[0,\infty )),\) \(\alpha ^2(x)+\beta ^2(x)\ne 0\), and N is the unite exterior normal vector to \(\partial \Omega\).
Throughout this paper, we always suppose that the following conditions hold:

(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 higherorder 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 \(\lambda _0\) 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 nonoscillatory 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 \(u(x,t)>0, \ u(x,t\tau )>0, \ u(x,\rho _k(t))>0,\) \(u(x,g(t,\xi ))>0\), \((x,t)\in \Omega \times [t_1,\infty ),k\in I_s.\)
Multiplying both sides of (1) by \(\varphi (x)\) and integrating with respect to x over the domain \(\Omega\), we have
From Green’s formula and boundary condition (2), it follows that
where \({\mathrm{d}}S\) is the surface element on \(\partial \Omega\).
If \(\alpha (x)\equiv 0, x\in \partial \Omega ,\) then from (2) we have
Hence, we obtain
If \(\alpha (x)\not \equiv 0, x\in \partial \Omega .\) Noting that \(\partial \Omega\) is piecewise smooth, \(\alpha , \ \beta \in C(\partial \Omega ,[0,\infty )), \alpha ^2(x)+\beta ^2(x)\ne 0\), without loss of generality, we can assume that \(\alpha (x)>0, \ x\in \partial \Omega .\) Then by (2) and (3) we have
Therefore, using Lemma 1, we obtain
Similarly, we have
It is easy to see that
Set
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.
(ii) Necessity. Suppose that Eq. (4) has a nonoscillatory solution \(\widetilde{V}(t)>0\). Without loss of generality we assume \(\widetilde{V}(t)>0\) for \(t\ge t_{*}\ge 0\), where \(t_{*}\) is some large number. From (4), we have
Multiplying both sides of (10) by \(\varphi (x)\), we obtain
Let \(\widetilde{u}(x,t)=\widetilde{V}(t)\varphi (x), \quad (x,t)\in \Omega \times [0,\infty )\). By Lemma 1, we have \(\Delta \varphi (x)=\lambda _0\varphi (x), \quad x\in \Omega\). Then (11) implies
which shows that \(\widetilde{u}(x,t)=\widetilde{V}(t)\varphi (x), \ (x,t)\in \Omega \times [t_*,\infty ),\) satisfies (1).
From Lemma 1, we get
which implies
Hence \(\widetilde{u}(x,t)=\widetilde{V}(t)\varphi (x)>0\) is a nonoscillatory 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
Suppose to the contrary that there is a nonoscillatory 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 \(u(x,t)>0, \ u(x,t\tau )>0, \ u(x,\rho _k(t))>0,\) \(u(x,g(t,\xi ))>0\), \((x,t)\in \Omega \times [t_1,\infty ),k\in I_s.\) As in the proof of Theorem 2, we obtain Eq. (9). By Lemma 1, from (9) we have
which shows that \(V(t)>0\) is a solution of the inequality (14). This is a contradiction. The proof of Theorem 4 is complete. \(\square\)
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
Consider the partial functional differential equation
with the boundary condition
Here \(n=6,\mu (t)=\frac{1}{5},\tau =\pi ,a(t)=3,a_1(t)=\frac{11}{5} ,\rho _1(t)=t\frac{3\pi }{2},p(t,\xi )=\frac{11}{5},g(t,\xi )=t+\xi ,\sigma (\xi )=\xi ,a=\pi ,b=\frac{\pi }{2}.\) It is easy to see that for \(t_0>0,\)
Then the conditions of Theorem 5 are fulfilled. Therefore every solution of the problem (19), (20) is oscillatory in \((0,\pi )\times [0,\infty )\). Indeed, \(u(x,t)=\sin x\cos t\) is such a solution.
Example 8
Consider the partial functional differential equation
with the boundary condition
Here \(n=4,\mu (t)=\frac{ 1}{ 2},\tau =\pi ,a(t)=\frac{ 1}{ 3},a_1(t)=\frac{ 1}{ 6} ,\rho _1(t)=t\frac{ \pi }{ 2},p(t,\xi )=\frac{ 1}{ 6},g(t,\xi )=t+\xi ,\sigma (\xi )=\xi ,a=\pi ,b=\frac{ \pi }{ 2}.\) It is easy to see that for \(t_0>0,\)
which shows that the conditions of Theorem 5 are satisfied. By Theorem 5, we obtain that every solution of the problem (21), (22) is oscillatory in \((0,\pi )\times [0,\infty )\). In fact, \(u(x,t)=e^{x}\cos t\) is such a solution.
Conclusions
This paper provides some oscillation criteria for solutions of higherorder 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.
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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.
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Li, W.N., Sheng, W. Oscillation of certain higherorder neutral partial functional differential equations. SpringerPlus 5, 459 (2016). https://doi.org/10.1186/s400640162111y
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DOI: https://doi.org/10.1186/s400640162111y