Abstract

New sufficient conditions are obtained for oscillation for the solutions of systems of a class of higher-order quasilinear partial functional differential equations with distributed deviating arguments. The obtained results are illustrated by example.

1. Introduction and Preliminary

In this paper, we consider systems of higher-order quasilinear partial functional differential equations with distributed deviating arguments: where is even, is a bounded domain in with a piecewise smooth boundary , and

Throughout this paper, we assume that the following conditions hold:(H1), , , ; , ;(H2), , , ;(H3), if , where , , , , and , ;(H4), , , , ;(H5), ; is nondecreasing functions and .

We consider two kinds of boundary conditions: where is the unit exterior normal vector to is a nonnegative continuous function on , and

During the past three decades, the investigation of oscillatory theory for partial functional differential equations has attracted attention of numerous researchers due to their significance in theory and applications. In 2000, Li [1] has investigated systems of second-order partial differential equations: and obtained sufficient conditions for the oscillation of solutions. In 2004, Lin [2] has studied higher-order nonlinear differential equation: and obtained sufficient conditions for the oscillation of solutions. In 2004, Li et al. [3] have investigated a class of vector hyperbolic differential equations with deviating arguments: and obtained sufficient conditions for oscillation of solutions. In 2009, Gui and Xu [4] have investigated even-order neutral type partial differential equations with distributed deviating arguments:

For related work, we refer the reader to [5–10]. However, to the best of our knowledge, few investigitions have been reported on the oscillation theory of systems of partial functional differential equations with distributed deviating arguments. Thus, the research presents its significance.

Definition 1. The vector function is said to be a solution of problem (1), (3) (or (1), (4)) if it satisfies (1) in and boundary condition (3) (or (4)).

Definition 2. A nontrivial component of the vector function is said to oscillate in if, for each , there is a point such that .

Definition 3. The vector solution of problem (1), (3) (or (1), (4)) is said to be oscillatory in if at least one of its nontrivial components is oscillatory in . Otherwise, the vector solution is said to be nonoscillatory.

2. The Main Results

Theorem 4. Assume that is a monotone decreasing function, and If there exist some such that then every solution of boundary value system (1), (3) is oscillatory.

Proof. Suppose (1) and (3) have a nonoscillatory solution . We assume that for , . Let , ; then , , . From (H1) and (H5), there exists a number such that , in . Multiplying both sides of (1) by and integrating with respect to over the domain , we get Therefore, From Green’s formula and boundary condition (3), it follows that where is the surface element on . Combining (12)-(13), we find Therefore, Set Then (15) yields Now, let , , ; from (17) we have Note that Similarly, we have Thus, from (18) for , we find It is clear that Therefore, it follows that Let We have , for . Hence, is a monotone decreasing function in the interval . We can claim that , for . In fact, if , for , then there exists a , such that . This implies that which contradicts the fact that . Now, we claim that . Otherwise . But from the well-known lemma of Kiguradze [5] and the fact that is even, the inequalities imply that higher derivatives of that function are also negative. This contradicts the fact that . Thus we have proved that is the increasing function of .
From (23), we find that there exists a such that Thus we obtain Since , from (27), we have Integrating inequality (28) from to , Furthermore, we have which contradicts condition (10). This proof is complete.

Theorem 5. Assume that is a monotone decreasing function and (9) holds. And
Then every solution of boundary value system (1), (3) is oscillatory.

Proof. As in the proof of Theorem 4, we obtain (23). Therefore, The remaining part of the proof is similar to that of Theorem 4; therefore it is omitted. The proof is complete.

Corollary 6. If inequality (23) has no eventually positive solution, then every solution of system (1), (3) is oscillatory in .

Theorem 7. Suppose that there exists a positive constant such that , and If there exist some such that then every solution of boundary value system (1), (3) is oscillatory.

Proof. As in the proof of Theorem 4, we obtain (27). Using the fact that , from (27), we have Since , from (35) we have Integrating inequality (36) from to , Furthermore, we have which contradicts condition (10). This proof is complete.

Theorem 8. Suppose that there exists a positive constant such that , and (33) holds. If then every solution of boundary value system (1), (3) is oscillatory.

Proof. As in the proof of Theorem 5, we obtain (32). The remaining part of the proof is similar to that of Theorem 7 and we omit it.
Consider the smallest eigenvalue of the Dirichlet problem: where is a positive constant and the corresponding eigenfunction is positive in .

Theorem 9. Assume all conditions of Theorem 4 hold. Then every solution of system (1), (4) is oscillatory in .

Proof. Suppose (1) and (4) have a nonoscillatory solution . We assume that for . Let ; then . From () and (), there exists a number such that , in . Multiplying both sides of (1) by , and integrating with respect to over the domain , we have Therefore, Green’s formula and boundary condition (4) yield Combining (42)-(43), we get Set Then (44) yields Let ; from (46) we have Now, as in the proof of Theorem 4, from (47) we have The remaining part of the proof is similar to that of Theorem 4; therefore it is omitted. The proof is complete.

Corollary 10. If inequality (48) has no eventually positive solution, then every solution of system (1), (4) is oscillatory in .

It is not difficult to see that the following theorems are true.

Theorem 11. Assume that is a monotone decreasing function, and (9) holds. If there exists some such that then every solution of boundary value system (1), (4) is oscillatory in .

Theorem 12. Suppose that there exists a positive constant such that , and (33) holds. If there exists some such that then every solution of boundary value system (1), (4) is oscillatory in .

Theorem 13. Assume all conditions of Theorem 5 hold. Then every solution of boundary value system (1), (4) is oscillatory in .

Theorem 14. Assume all conditions of Theorem 7 hold. Then every solution of boundary value system (1), (4) is oscillatory in .

Theorem 15. Assume all conditions of Theorem 8 hold. Then every solution of boundary value system (1), (4) is oscillatory in .

3. Example

Example 1. Consider the systems of second-order partial differential equations with boundary condition
Here, , , , , , , , , , , , , , , , , , . It is easy to see that all conditions of Theorem 4 are fulfilled. Then every solution of system (52), (53) oscillates in . In fact, . is such a solution.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors thank referees for both a careful reading of the paper and useful suggestions that helped to improve the presentation. This research is supported by Natural Sciences Foundation of China (no. 11172194) and Natural Sciences Foundation of Shanxi Province (no. 2010011008).