Department of Mathematics, Zhanjiang Normal University, Zhanjiang, Guangdong, 524048, China

Abstract

This paper investigates integral inequalities with delay for discontinuous functions involving two nonlinear terms. We do not require the classes ℘ and

**MSC: **
26D15, 26D20.

1 Introduction

The Gronwall-Bellman integral inequalities and their various linear and nonlinear generalizations, involving continuous or discontinuous functions, play very important roles in investigating different qualitative characteristics of solutions for differential equations and impulsive differential equations such as existence, uniqueness, continuation, boundedness, continuous dependence of parameters, stability, attraction, practical stability. The literature on inequalities for continuous functions and their applications is vast (see

about the nonnegative piecewise continuous function

He replaced the constant

In 2005, he

In 2007, Iovane

Later, Gallo and Piccirillo

with a general nonlinear term

Motivated by this observation, in this paper, we consider the following much more general inequality

with two nonlinear terms

2 Main results

Consider (1.7), and assume that

(C_{1})

(C_{2})

(C_{3})

(C_{4})

(C_{5}) For

Let

**Theorem 2.1** _{k}) (

The proof is given in Section 3.

**Remark 2.1** (1) If

(2) Take

with

Hence,

After recursive calculations, we have for

which is same as the one in

(3) Clearly, (1.2) and (1.3) are special cases of (1.7). If

Let

Consider the inequality

which looks more complicated than (1.7).

**Corollary 2.1** _{1})-(C_{3}) _{5})

where

which is just the form of (1.7), if we take

□

**Remark 2.2** Using the same way, we can change inequality (1.4) into the form of (1.7) with

3 Proof of Theorem 2.1

Obviously,

We first consider

Take any fixed

for

and

Integrating both sides of the inequality above, from

for

where

It is easy to check that _{1}), we have by (2.3)

Note that

Integrating both sides of inequality (3.3), from

Thus,

We have by (2.3)

Since the inequality above is true for any

Replacing

This means that (2.1) is true for

For

where the definition of

This implies that (2.1) is true for

Assume that (2.1) is true for

for

For

where we use the fact that the estimate of

This yields that (2.1) is true for

4 Applications

Consider the following impulsive differential equation

where

Assume that

(1)

(2)

The solution of (4.1) with an initial value

which implies that

Let

so (4.3) is same as (1.7). It is easy to obtain for any positive constants

Thus, for any nonnegative integer

provided that

**Remark 4.1** From (4.3), we know that

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All the authors read and approved the final manuscript.

Acknowledgements

This work was supported by the Project of Department of Education of Guangdong Province, China (No. 2012KJCX0074), the PhD Start-up Fund of the Natural Science Foundation of Guangdong Province, China (No. S2011040000464), the China Postdoctoral Science Foundation-Special Project (No. 201104077), the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry (No. (2012)940), the Natural Fund of Zhanjiang Normal University (No. LZL1101), and the Doctoral Project of Zhanjiang Normal University (No. ZL1109).