By using the properties of modified Riemann-Liouville fractional derivative, some new delay integral inequalities have been studied. First, we offered explicit bounds for the unknown functions, then we applied the results to the research concerning the boundness, uniqueness and continuous dependence on the initial for solutions to certain fractional differential equations.
The common differential and integral inequalities are playing an important role in the qualitative analysis of differential equations. At the same time, delay integral and differential inequality have been studied due to their wide applications [
In 2008, Zhiling Yuan, et al. [
then they offered an explicit estimate for
In 2013, Bin Zheng and Qinghua Feng [
and they applied the obtained results to study the properties of solution
In this paper, combining (1) and (2), we will explore the following form of delay integral inequality
Now we list some Definitions and Lemmas which can be used in this paper.
Definition 1. [
Definition 2. [
Some important properties for the modified Riemann-Liouville derivative and fractional integral are listed as follows [
(1)
(2)
(3)
(4)
(5)
Lemma 1. [
Lemma 2. [
Then for
Implies
Theorem 1 Assume that
with the initial condition
where
for any
Proof. Fix
Since
and
so we have
we have
and
So for
for
Combining (10) and (11), we obtain
From (8), (9) and (12) we get
By Lemma 1 we have
Since
for
so we have
Using Lemma 2 to (14) we get
Letting
Combining (8) and (17), we get (6).
Remark 1. Assume
Theorem 2. Assume that
where
with the condition (5) in Theorem 1, then we have
where
Proof. Let
Since
for
so we can get
and
By Lemma 1 we get for any
Proceeding the similar proof of Theorem 3 in [
From (23), (24), (25) and condition (19) we have
By Lemma 2 we have
Combining (22) and (27), (20) can be obtained subsequently.
Theorem 3. Assume that
then
where
Proof. Let
then we get
Since
stant
so we get
By Lemma 2 we have
Combining (30) and (31), we get (29).
Remark 2. Considering
Theorem 4. Assume that
then
where
Proof. Let
then we get
The assumptions on
Then we get
so we have
and
Using Lemma 4 to (35), we can get
Combining (34) and (36), we get (33).
Remark 3. Considering
In this section, we will show that the inequalities established above are useful in the research concerning the boundness, uniqueness and continuous dependence on the initial value for solutions to fractional differential equations.
with the condition
where
And
Example 1. Assume that
where
where
Proof. By Equation (37), we have
By (39) and (41) we can get
With a suitable application of Theorems 1 to (42) (with
Example 2. Assume that
where
Proof. Suppose
Furthermore,
which implies
Through a suitable application of Theorem 1 to (44) (with
which implies
Example 3. Suppose that
If
Proof. By Equation (45), we have
so we get
Furthermore
Apply Theorem 1 to (47) (with
where
Example 4. Assume that
Proof. By Equation (48) we can get
with a suitable application of Theorem 3 to (49) (with
where we used
Example 5. If
Proof. Suppose
Furthermore,
which implies
With a suitable application of 3 to (50) (with
which implies
Example 6. Suppose that
Then all the solutions of Equation (48) depend on the initial value
Proof. By Equation (51), we have
so we get
Furthermore
Apply Theorem 3 to (52) (with
where we use the fact that
This gives that
We thank the Editor and the referee for their comments. This work is supported by National Science Foundation of China (11171178 and 11271225).