1. Introduction
The fractional differential equations are used to describe mathematical models of numerous real processes and phenomena studied in many areas of science and engineering such as population dynamics, neural networks, industrial robotics, electric circuits, optimal control, biotechnology, economics and many other branches of science. Furthermore, the fractional calculus can also provide an excellent instrument for the description of memory and hereditary properties of various materials and processes due to the existence of a “memory” term in the model.
The oscillation theory as a part of the qualitative theory of differential equations has been developed rapidly in the last decades, and there has been a great deal of works on the oscillatory behavior of integer order differential equations [1] [2] [3]. As a new cross-cutting area, recently some attention has been paid to oscillations of fractional differential equations [4] [5] [6] [7]. Some new developments in the oscillatory behavior of solutions of fractional differential equations with damping terms [8] [9] [10] [11] have been reported by authors.
In this paper, we consider the oscillatory behavior of solutions of the following fractional differential equation:
(1.1)
where
,
is the Laplacian in
,
is a bounded domain in
with a piecewise smooth boundary
.
is a real number and
is the Riemann-Liouville left-sided fractional derivative of order
of
for
.
We shall consider Robin and Dirichlet boundary conditions
(1.2)
and
(1.3)
where
is continuous function,
is the unit out normal vector to
.
The following conditions are assumed to hold:
(H1)
are convex in
and
is a monotone increasing function with
for
, there exist positive constants
,
such that
for
. And
,
,
,
,
,
,
,
.
(H2)
and
are positive continuous functions on
for a certain
, and
is a nonpositive continuous function on
for a certain
. There exists a constant
,
for
. And
,
,
.
(H3)
is continuous function with
for
, there exists positive constant
such that
for
, and
for
.
(H4)
, and
.
By a solution of (1.1), (1.2) and (1.3), it mean a nontrivial function
with
,
satisfies (1.1) for
on
and the boundary conditions (1.2) and (1.3).
A solution
of (1.1) is said to be oscillatory if it is neither eventually positive nor eventually negative, otherwise it is nonoscillatory. Equation (1.1) is said to be oscillatory if all its solutions are oscillatory.
2. Preliminaries
In this section, there are several kinds of definitions of fractional derivatives and integrals and some lemmas which are useful throughout this paper.
Definition 2.1 [12] The Liouville left-sided fractional integral of order
of a function
on the half-axis
is given by
provided that the left side is pointwise defined on
, where
is the gamma function.
Definition 2.2 [12] The Riemann-Liouville fractional partial derivative of order
with respect to t of a function
is given by
Lemma 2.3 [12] Let
Then
Lemma 2.4 [4] If
and
are nonnegative, then
where the equality holds if and only if
.
3. Oscillation of (1.1) and (1.2)
For the sake of convenience, we set
Theorem 3.1 Suppose that (H1)?(H5) hold and if the fractional differential inequality
(3.1)
has no eventually positive solution and the fractional differential inequality
(3.2)
has no eventually negative solution, every solution of (1.1) and (1.2) is oscillatory in
.
Proof Suppose that
is a nonoscillatory solution of (1.1) and (1.2), it is either eventually positive or eventually negative. Without loss of generality, we may assume that
is an eventually positive solution of (1.1) and (1.2) in
. Integrating (1.1) with respect to
over
, we obtain
(3.3)
Using Green’s formula and boundary condition (1.2), it is obvious that
(3.4)
(3.5)
By using Jensen’s inequality and (H1), (H4), we get
(3.6)
Combining (3.3)?(3.6), we obtain
(3.7)
Therefore,
is an eventually positive solution of (3.1), this contradicts the hypothesis.
Secondly, if
is an eventually negative solution of the problem (1.1) and (1.2), then using above procedure, we can easily show that
is an eventually negative solution of the Equation (3.2).
This completes the proof.
Theorem 3.2 Suppose that (H1)?(H4) and
(3.8)
hold. if there exists a positive function
such that
(3.9)
where
,
are defined as in (H1), then every solution of (3.1) and (3.2) is oscillatory.
Proof Suppose that
is a nonoscillatory solution of (3.1). Without loss of generality, we may assume that
is an eventually positive solution of (3.1). Then there exists
, where
is defined as in Lemma 2.3.
It follows from (3.7) that
(3.10)
Thus,
is strictly decreasing on
. Since
for
and (H1), we see that
is eventually of one sign. We claim that
(3.11)
If not, there exists
such that
. Since
is strictly decreasing on
and it is clear that
, where
is a constant for
. Therefore, we have
(3.12)
Due to
and
, we get
(3.13)
Integrating the above inequality from
to
, from Lemma 2.3, we have
(3.14)
which yields
(3.15)
By (H2) and (3.8), letting
, we get
. This contradicts the fact that
. Hence, (3.11) holds.
From Lemma 2.3
(3.16)
therefore,
(3.17)
Define the function
by the generalized Riccati substitution
(3.18)
Then we have
for
, and from (3.18), it follows that
(3.19)
Taking
(3.20)
from Lemma 2.4 and (3.19), we get
(3.21)
Integrating both sides of the inequality (3.21) from
to
, and taking the limit supremum of both sides of the above inequality as
, we get
(3.22)
Which contradicts (3.9). The proof is complete.
Secondly, if
is an eventually negative solution of the fractional differential inequality (3.2) and there exists
. When (3.2) is oscillatory is similar to that of above procedure, and hence is omitted.
Theorem 3.3 Assume that (H1) - (H4) and (3.8) hold. Furthermore, suppose that there exist a positive function
and a function
, where
, such that
where
and
has a nonpositive continuous partial derivative
with respect to the second variable and satisfies
(3.23)
where
,
and
are defined as in Theorem 3.2. Then all solutions of (3.1) and (3.2) are oscillatory.
Proof Suppose that
is a nonoscillatory solution of (3.1). Without loss of generality, we may assume that
is an eventually positive solution of (3.1). We proceed as in the proof of Theorem 3.2 to get (3.21), Multiplying (3.21) by
and integrating from
to
, for
, we derive
(3.24)
Therefore,
(3.25)
which is a contradiction to (3.23). The proof is complete.
Secondly, if
is an eventually negative solution of the fractional differential inequality (3.2). The proof when (3.2) is oscillatory is similar to that of above procedure, and hence is omitted.
Next, we consider the case
(3.26)
which is different from (3.8). In this case, we have the following results.
Theorem 3.4 Assume that (H1)?(H4) and (3.26) hold, and that there exist a positive function
such that (3.9) holds. If for every constant
, such that
(3.27)
Then every solutions
of (3.1) and (3.2) are oscillatory or satisfies
or
, where
is defined as Lemma 2.3.
Proof Suppose that
is a nonoscillatory solution of (3.1). We may assume that
is an eventually positive solution of (3.1), proceeding as in the proof of Theorem 3.2 to get (3.10). Then there are two cases for the sign of
.
When
is eventually positive is similar to that of Theorem 3.2, we get that every solution
of (3.1) is oscillatory.
If is eventually negative, there exists, such that for. From (3.16), therefore,
(3.28)
Since and holds, then we obtain
(3.29)
Letting in (3.28), we have
(3.30)
If, then there exists such that for. Thus, we get and,. Now we claim that. If not, that is, then from (3.10), we derive
(3.31)
Integrating both sides of (3.31) from to, we have
(3.32)
where, Hence, from (H2) and (3.32), we get
(3.33)
Integrating both sides of (3.33) from to, we obtain
(3.34)
using and (3.27), as,. which contradicts the fact that. Therefore, we have, that is. The proof is complete.
Secondly, if is an eventually negative solution of the fractional differential inequality (3.2). The proof when (3.2) is oscillatory is similar to that of above procedure, and hence is omitted.
Theorem 3.5 Assume that (H1) - (H4) and (3.26) hold, Let and be defined as in Theorem 3.3 such that (3.23) holds. Furthermore, assume that (3.27) holds for every. Then every solutions of (3.1) and (3.2) are oscillatory or satisfies or, where is defined as Lemma 2.3.
Proof Suppose that is a nonoscillatory solution of (3.1). Without loss of generality, assume that is an eventually positive solution of (3.1), and proceeding as in the proof of Theorem 3.2 to get (3.11), there are two cases for the sign of.
When is eventually positive, the proof is similar to that of Theorem 3.3. is eventually negative, the proof is similar to that of Theorem 3.4. Here we omitted it.
4. Oscillation of (1.1) and (1.3)
In the next we establish sufficient conditions for the oscillation of all solutions of (1.1), (1.3). For this we need the following:
The smallest eigenvalue of the Dirichlet problem
(4.1)
is positive and the corresponding eigenfunction is positive in.
Theorem 4.1 Let all the conditions of Theorem 3.2 and 3.3 be hold. Then every solution of (1.1) and (1.3) oscillates in.
Proof Suppose that is a nonoscillatory solution of (1.1) and (1.3). Without loss of generality, we may assume that is an eventually positive solution of (1.1) and (1.3) in for. Multiplying both sides of the Equation (1.1) by and then integrating with respect to over, we obtain for,
(4.2)
Using Green's formula and boundary condition (1.3), it is obvious that
(4.3)
(4.4)
By using Jensen's inequality and (H1) and (H4), we get
(4.5)
Set
Combining (4.2)-(4.5), we obtain
(4.6)
The rest of the proof is similar to that of Theorems 3.2 and 3.3, and hence the details are omitted.
Theorem 4.1 Let the conditions of Theorem 3.4 hold. Then every solution of (1.1) and (1.3) is oscillatory or satisfies or, where is defined as Lemma 2.3.
Theorem 4.2 Let the conditions of Theorem 3.5 hold; Then every solution of (4.6) is oscillatory or satisfies or, where is defined as Lemma 2.3.
The proofs of Theorem 4.1 and 4.2 are similar to that of Theorems 3.2-3.5 and hence the details are omitted.
5. Applications
Example 1 Consider the fractional differential equation
(5.5)
Here, , , , , and.
Taking, , , , , , ,.
Then, we get
It is clear that conditions (H1) - (H4) and (3.1) hold. Furthermore, taking
,
which satisfies condition (3.10). For every constant, , we obtain
Which shows that (3.27) holds. Therefore, by Theorem 3.4 every solution of (5.5) is oscillatory or satisfies or.
Acknowledgements
This research was partially supported by grants from the National Basic Research Program of China, No. 41630643 and by the Science Foundation for The Excellent Youth Scholars of Ministry of Education of China, No. 11801530.