Forced Oscillation Criteria for a Class of Fractional Partial Differential Equations with Damping Term

Abstract

Sufficient conditions are established for the forced oscillation of fractional partial differential equations with damping term of the form , , with one of the two following boundary conditions: or , where is a bounded domain in with a piecewise smooth boundary, ,   is a constant, is the Riemann-Liouville fractional derivative of order of with respect to , is the Laplacian in , is the unit exterior normal vector to , and is a continuous function on . The main results are illustrated by some examples.

1. Introduction

Fractional differential equations are generalizations of classical differential equations to an arbitrary (noninteger) order and have gained increasing attention because of their varied applications in various fields of applied sciences and engineering. In the past few years, the theory of fractional differential equations and their applications have been investigated extensively; for example, see the monographs [1–4].

Recently, the oscillatory behavior of solutions for fractional differential equations was discussed in [5–13] and so forth. In [13], the authors studied the oscillation of solutions to nonlinear fractional differential equations of this form with initial condition , where is a real number, is a constant, and is the Riemann-Liouville fractional derivative of order of .

However, to the best of author’s knowledge very little is known regarding the oscillatory behavior of fractional partial differential equations up to now; we refer to [14–16]. In particular, nothing is known regarding the oscillation properties of the problems (1)-(2) and (1)–(3) up to now.

In this paper we investigate the forced oscillation of fractional partial differential equations with damping term of the form where is a bounded domain in with a piecewise smooth boundary , ,   is a constant, is the Riemann-Liouville fractional derivative of order of with respect to , and is the Laplacian in .

We assume throughout this paper that(A1), ;(A2), and ;(A3).

Consider one of the two following boundary conditions: or where is the unit exterior normal vector to and is a continuous function on .

By a solution of the problems (1)-(2) (or (1)–(3)), we mean a function which satisfies (1) on and the boundary condition (2) (or (3)).

A solution of the problems (1)-(2) (or (1)–(3)) is said to be oscillatory in if it is neither eventually positive nor eventually negative, otherwise it is nonoscillatory.

Definition 1. The Riemann-Liouville fractional partial derivative of order with respect to of a function is given by provided the right hand side is pointwise defined on , where is the gamma function.

Definition 2. The Riemann-Liouville fractional integral of order of a function on the half-axis is given by provided the right hand side is pointwise defined on .

Definition 3. The Riemann-Liouville fractional derivative of order of a function on the half-axis is given by provided the right hand side is pointwise defined on , where is the ceiling function of .

2. Main Results

In this section, we establish the oscillation of the problems (1)-(2) and (1)–(3). We firstly introduce the following lemmas which are very useful in the proof of our main results.

Lemma 4 (see [17]). The smallest eigenvalue of the Dirichlet problem, is positive and the corresponding eigenfunction is positive in .

Lemma 5 (see [2, page 74]). Let , , and . If the fractional derivatives and exist, then

Lemma 6 (see [2, page 75]). Let and be the fractional integral (5) of order , then

For convenience, one uses the following notations in this paper: where is the surface element on .

Theorem 7. Assume that where is a constant. If then every solution of the problems (1)-(2) is oscillatory in , where and is a constant.

Proof. Suppose to the contrary that there is a nonoscillatory solution of the problems (1)-(2) which has no zero in for some . Then or for .
Case 1 (, ). Integrating (1) with respect to over the domain , we have Green’s formula and (2) yield From (A2), it is easy to see that By Lemma 5, it follows from (13)–(15) that From (16), we easily see that Integrating both sides of the above inequality from to , we obtain where . Using Lemma 6, it follows from (18) that Taking in (19), we have which contradicts the fact that .
Case 2 (, ). Using the procedure of the proof of Case 1, we conclude that (13) and (14) are satisfied. From (A2), we have Combining (13), (14), and (21), we have It follows from (22) that Integrating both sides of the above inequality from to , we obtain Using Lemma 6, from (24) we obtain Taking in (25), we have which contradicts the fact that . This completes the proof of Theorem 7.

Theorem 8. Assume that where is a constant. If then every solution of the problems (1)–(3) is oscillatory in , where and is a constant.

Proof. Suppose to the contrary that there is a nonoscillatory solution of the problems (1)–(3) which has no zero in for some . Then or for .
Case 1 (, ). Multiplying both sides of (1) by and integrating with respect to over the domain , we have Green’s formula and (3) yield From (A2), it is easy to see that Combining (29)–(31), we easily see that The remainder of the proof is similar to that of Case 1 of Theorem 7 and we can obtain a contradiction to .
Case 2 (,  ). Using the procedure of the proof of Case 1, we conclude that (29) and (30) are satisfied. From (A2), we have Combining (29), (30), and (33), we obtain Using a similar way in the proof of Case 2 of Theorem 7, we can obtain a contradiction to . The proof of Theorem 8 is complete.

3. Examples

Example 1. Consider the following fractional partial differential equation: with the boundary condition Here , , , , , , and  . It is obvious that , , , and Hence Letting , we obtain Setting , above integral (39) can be written as the following form: Noting that we obtain that are convergent.
Therefore, combining (39)–(41) and noting the fact that (42) are convergent, by careful calculation, we can get which shows that all the conditions of Theorem 7 are fulfilled. Then every solution of the problems (35)-(36) oscillates in .

Example 2. Consider the following fractional partial differential equation: with the boundary condition Here , ,  ,  , , , and .
It is obvious that ,  ,  , and  . Therefore, Hence Using a similar way in Example 1, we can obtain Therefore, by Theorem 8, it is easy to see that every solution of the problems (44)-(45) is oscillatory in .

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (10971018). The author thanks the referee for his valuable comments and suggestions on this paper.