Oscillation Theorems for Two Classes of Fractional Neutral Differential Equations ()
1. Introduction
The fractional differential equation (FDE) has gained considerable importance due to its various application in fluid mechanics, viscoelasticity, electrochemistry of corrosion, classical mechanics and particle physics, control theory, diffusive systems, and so on [1] - [7]. Therefore, the theory of fractional calculus has received extensive attention from scholars at home and abroad. In the past few decades, many researchers have done a lot of research on the properties of fractional differential, for example, the existence, uniqueness, stability, asymptoticity of solutions of fractional differential equations and numerical solutions of fractional differential equations, etc. [8] - [13].
However, to the best of our knowledge, there are few results on oscillation for the fractional differential equation. We refer to [14] - [19] and the references therein. In [20], Meng et al. studied the linear fractional order delay differential equation
where
,
,
.
In [21], A. George Maria Selvam and R. Janagaraj establish oscillation theorems for damped fractional order differential equation of the form:
where
and
defined as the difference operator of the Riemann-Liouville derivative of order
.
In [22], Zhu et al. studied forced oscillatory properties of solutions to nonlinear fractional differential equations with damping term and time delay:
where
when
and
is a given continuous function, where
and
are constants,
, b is a real number,
is the Riemann-Liouville fractional derivative of order
of y.
In [23], Zhou et al. study the following fractional functional partial differential equation involving Riemann-Liouville fractional derivative:
supplemented with the initial condition
and boundary conditions
Motivated by the analysis above, in this paper, we are concerned with the oscillation of two classes of fractional differential equations as follows:
(1)
(2)
where
,
is Riemann-Liouville fractional derivative of order
.
This paper is organized as follows. In the next section, we introduce some useful preliminaries. In Section 3, we present various sufficient conditions for the oscillation of all solutions to Equations (1) and (2) by using fractional calculus, Laplace transforms and Green’s function. Finally, we provide some examples to show the applications of our criteria.
2. Preliminaries
In this section, we introduce preliminary facts which are used throughout this paper.
Definition 1 ( [24]). Let
be a finite interval and let
be the space of functions f which are absolutely continuous on
. It is known (see [25], p.338) that
coincides with the space of primitives of Lebesgue summable function:
Definition 2 ( [24]). The Riemann-Liouville left-sided fractional integral of order
of a function
on the half-axis
is given by:
provided the left hand side is pointwise defined on
, where
is the gamma function.
Definition 3 ( [24]). The Riemann-Liouville left-sided fractional derivative of order
of a function
on the half-axis
is given by:
provided the left hand side is pointwise defined on
.
We recall some facts about Laplace transforms. If
is the Laplace transform of
,
then the abscissa of convergence of
is defined by:
Therefore,
exists for
.
Definition 4. A function
is eventually positive if there is a
such that
for all
, where
.
Definition 5. By a solution of (3) in
with the initial function
, we mean a function
such that
,
,
exists and
satisfies (3) in
. A nontrivial solution
of equation (3) is said to oscillate if it has an arbitrarily large number of zeros. Otherwise, the solution is called non-oscillatory.
Lemma 1 ( [24]). Let
be the Laplace transform of the Riemann-Liouville fractional derivative of order
with the lower limit zero for a function x, and
is the Laplace transform of
. Further, for
and for any
,
holds for constant
and
. Then the relation
is valid for
.
Lemma 2 ( [23]). For any
, the Laplace transform
of
exists and has the same abscissa of convergence as
.
Lemma 3 ( [26]). Let
be continuous function. If w is non-decreasing and there are constants
and
such that
then there exists a constant
with
for every
.
3. Main Results
In this section, we present our main results.
Lemma 4.
is the Riemann-liouville derivative of order
with the lower limit zero for a function
,
is the Laplace transform of
,
,
, then the following relation holds.
Proof.
By Lemma 2,
exists such that
then
The proof is complete. □
We consider the following fractional-order delay differential equation:
(3)
where
,
. The standard initial condition associated with (3) is
(4)
where
,
.
Lemma 5. If
, the solution of Equation (3) has an exponent estimate
for constants
.
Proof. Taking the Riemann-Liouville integral of Equation (3), we have
(5)
where
,
.
As
, there exists a constant
such that
. For
, we have
(6)
which, together with (5), yields
(7)
In the interval
, taking the maximum value on both sides of inequality (7), then
One can introduce nondecreasing function
as
then we have
By lemma 3, there exists a constant
such that
Obviously, from the above formula, we infer that
has an exponent estimate. The proof is complete. □
Theorem 1. Assume that
,
. If the equation
(8)
has no real roots, then every solution of (3) is oscillatory.
Proof. Suppose that
is a nonoscillatory solution of Equation (3). Without loss of generality, we assume that
is an eventually positive solution of Equation (3) which means that there exists a constant T such that
for
. Since Equation (3) is autonomous, we may assume that
for
. Taking Laplace transform of both sides of (3), we obtain
where
,
.
Hence
(9)
Let
then, from (9), we get
(10)
Since
has no real roots and
,
. By positivity of
in
, then exists constant
such that
. Since
,
then
Thus there exists a constant
such that for
,
Then,
Thus we conclude that
, but
and
are positive. Hence, (10) leads to a contradiction. The proof is complete. □
Corollary 1. Assume that
,
. If the equation
has no real roots, then every solution of (3) is oscillatory.
Proof. If we modify the function
and
, defined in Theorem 1, as
Then, following the method of proof for Theorem 1, one can complete the proof.
□
Corollary 2. Assume that
,
. If the equation
has no real roots, then every solution of (3) is oscillatory.
Proof. Proceeding as in the proof of Theorem 1, according to (9) and (10), for
, we have
Thus, we obtain
Thus we conclude that
, but
and
are positive. Hence (10) leads to a contradiction. The proof is complete. □
Theorem 2. Assume that
,
, and if
(11)
then every solution of Equation (3) is oscillatory.
Proof. Assume that Equation (8) has a real roots
, if
, then
, it is impossible. Thus we conclude that
. Since
is the ratio of two odd integers, it follows from (8) that
then
(12)
By (12) and the inequality
for
, we get
which implies that
this is a contradiction. The proof is complete. □
Theorem 3. Assume that
,
, and if
(13)
then every solution of Equation (3) is oscillatory.
Proof. In regard to
, since
for
, thus we conclude that if
has real roots, the roots are definitely less than zero.
Let
(14)
from (14), we get
(15)
Equation (15) has only one real root for
. Thus (14) has only one real root for
, and Assume that the root is
. Meanwhile, as for
,
, thus
has only one extreme point, which is also the minimum point for
.
Let
(16)
since
and
, we can obtain
(17)
By
and (14), we get
and by (17) and (13), we obtain
Thus we conclude
, and then
. By Corollary 1, every solution of (3) is oscillatory. The proof is complete. □
Theorem 4. Assume that
,
, if
(18)
then every solution of Equation (3) is oscillatory.
Proof. To prove the result, it suffices to prove that (8) has no real roots under the conditions (18). One can note that any real root of (8) cannot be positive and since
,
is not a root. Thus any real root of (8) can only be negative if it is possible. Let us set
for convenience and show that
(19)
From (19), we have
(20)
Let
Then
So, we get
This contradicts (18) which implies that (8) has no real roots. By Corollary 2, then every solution of (3) is oscillatory. The proof is complete. □
Now, we consider fractional differential equations with multiple delays
(21)
where
,
.
Lemma 6. If
,
, then the solution of Equation (21) has an exponent estimate
for constants
.
Proof. The proof of this conclusion is similar to that of Lemma 5, and thus we omit it. □
Theorem 5. Assume that
,
,
, if
(22)
has no real roots, then every solution of (21) is oscillatory.
Proof. Taking Laplace transform of both sides of (21), we obtain
where
,
.
Hence,
(23)
Let
then, from (23), we get
(24)
Then, the following proof process is similar to Theorem 1. one can complete the proof. □
Corollary 3. Assume that
,
,
, if
(25)
then every solution of (21) is oscillatory.
Proof. By using the arithmetic-geometric mean inequality
for
, we find
(26)
Let
(27)
where
,
,
,
and
.
By (25) and the proof process of Theorem 2, we can obtain
for
. This is, (22) has no real roots, by Theorem 4, every solution of (21) is oscillatory. The proof is complete. □
We consider the following fractional order delay differential equation
(28)
(H):
,
, for
,
,
.
The standard initial condition associated with (28) is
(29)
where
,
.
Theorem 6. Assume that (H) holds, and if
(30)
then every solution of Equations (28) is oscillatory.
Proof. Assume that (28) has an eventually positive solution
, that is, there exists a sufficiently large positive constant T such that
,
,
for
, from (28), we obtain
Then we can see that the eventually positive solution
satisfies the inequality
(31)
According to (30) and Theorem 2, it follows that Equation
has no real roots. Therefore, similarly to the proof of Theorem 1, inequality (31) has no eventually positive solution, which implies that every solution of (28) oscillates. □
We consider the delay fractional order partial differential equation
(32)
where
. Here
is a bounded domain with boundary
smooth enough. The hypotheses are always true as follows:
(H1):
;
;
.
(H2):
,
, for
.
The initial condition
,
,
. Consider the boundary conditions as follows:
(33)
where N is the unit exterior normal vector in
.
Theorem 7. Assume that
,
,
, if
then every solution of (32) with the boundary condition (33) is oscillatory.
Proof. Assume that (32) with the boundary condition (33) has no oscillation solution, without loss of generality, we assume that
is an eventually positive solution of (32) which implies that there exists
such that
,
,
in
. Since (H1) and (H2), from (32), we can obtain
(34)
Integrating (34) with respect to x over
yields
(35)
By Green’s formula and the boundary condition (33), we have
(36)
Let
, from (35) and (36), we get
(37)
That is, there exists eventually positive solution for inequality (37). According to the conditions of Theorem 7 and Corollary 3, it follows that Equation
has no real roots. Therefore, similarly to the proof of Theorem 5, inequality (37) has no eventually positive solution which implies that every solution of (32) oscillates. □
4. Example
Example 1. Consider the following fractional differential equation
(38)
Notice
,
,
,
,
, then it is easy to find
and
, then (38) is oscillatory by Theorem 6.
Example 2. Consider the following fractional differential equation
(39)
with the boundary conditions
(40)
Notice
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
, then it is easy to find
,
,
,
.
Therefore,
then (39) is oscillatory by Theorem 7.
Acknowledgements
This research was partially supported by grants from the National Natural Science Foundation of China, No. 41630643.