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 [1] -[3] . In recent years, the fractional differential and fractional integrals are adopted in var- ious fields of science and engineering. In addition, the fractional differential inequalities have also been studied [4] -[10] . We also need to study the delay differential equation and delay differential inequalities when dealing with certain problems. However, to the best of our knowledge, very little is known regarding this problem [11] . In this paper, we will investigate some delay integral inequalities.

In 2008, Zhiling Yuan, et al. [3] studied the following form delay integral inequality

then they offered an explicit estimate for, and applied this result to research the properties of solution to certain differential equations.

In 2013, Bin Zheng and Qinghua Feng [6] put forward the following form of fractional integral inequality

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. [6] The modified Riemann-Liouville derivative of order is defined by

Definition 2. [6] The Riemann-Liouville fractional integral of order on the interval is defined by

.

Some important properties for the modified Riemann-Liouville derivative and fractional integral are listed as follows [6] (the interval concerned below is always defined by).

(1),

(2),

(3),

(4),

(5).

Lemma 1. [3] Assume that, , and, then

.

Lemma 2. [6] Let, , , be continuous functions defined on.

Then for,

Implies

.

Theorem 1 Assume that, , , , , , , and, are nondecreasing functions in. If satisfies the following form of delay integral inequality

with the initial condition

,

where, , , , , , , are constants, , ,

and, then we have

for any, where

Proof. Fix, let

,

Since, , , , , , , there’s exist a constant, such that

,

and is convergence integral,

so we have

we have is a nonnegative and nondecreasing. From (4) and (7) we get

and

So for with, we have

for with, we have

Combining (10) and (11), we obtain

From (8), (9) and (12) we get

By Lemma 1 we have

Since, , are continuous and there exists a constant satisfies

for, where. Then we get

,

so we have

Using Lemma 2 to (14) we get

Letting in (16) and considering is arbitary, after substituting with, we get

Combining (8) and (17), we get (6).

Remark 1. Assume, then the inequalities in Theorem 1 reduce to Lemma 5 in [6] .

Theorem 2. Assume that, , , , , , , are defined as in Theorem 1. If satisfies the following form of integral inequality,

where, , , are constants and satisfy

with the condition (5) in Theorem 1, then we have

where

.

Proof. Let

Since are nonegative functions, is also nonegative and nondecreasing func- tion, in addition, there exists a constant satisfying

for, where. Then we get

,

so we can get. From (18) we have

and

By Lemma 1 we get for any,

Proceeding the similar proof of Theorem 3 in [3] , we can get

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, , , , , and is nondecreasing with for. If satisfies

then

where

.

Proof. Let

,

then we get

Since are both continuous functions, is continuous and there exists a con-

stant satisfying for, where. Then we get

,

so we get and

By Lemma 2 we have

Combining (30) and (31), we get (29).

Remark 2. Considering in Theorem 3, proceeding the similar proof of Theorem 3, we can get

.

Theorem 4. Assume that, , , , and is nonde- creasing with for, with. If satisfies the following form of delay integral inequality

then

where

,

.

Proof. Let

,

then we get

The assumptions on and imply that is nondecreasing and there exists a constant

satisfying for, where.

Then we get

,

so we have. For, we have

,

and

Using Lemma 4 to (35), we can get

Combining (34) and (36), we get (33).

Remark 3. Considering and in Theorem 4, we can get Remark 2.

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.

We thank the Editor and the referee for their comments. This work is supported by National Science Foundation of China (11171178 and 11271225).