We investigate the following differential equations:

The theory of impulsive differential equations in abstract spaces has become a new important branch and has developed rapidly (see [

In this paper, we study the existence of positive solutions for the following system:

When

Let

Throughout the rest of the paper, we assume

(H1)

This paper aims to obtain the positive solution for (

Let

For two differential functions

Consider the linear nonhomogeneous problem of the form

Suppose that

We just need to show that the function

Consider the following boundary value problem with integral boundary conditions:

(H2) Let

For convenience, we denote

Let

We solve (

Evidently, the statement of Lemma

For the solution

By Lemma

From (

Let condition (H1) hold. Then for the Wronskian of solution

Using the initial conditions (

From (

Under condition (H1) the Green's function

Let

Let us set

Define a mapping

The fixed point of the mapping

Clearly,

Let

(i) For for all

(ii) If

Defined a linear operator

If (H2) is satisfied, then

(i)

Using

Let

(ii) We want to show that

Since

So

Conversely, 1 is not an eigenvalue of

(iii) We use the theory of Fredholm integral equations to find the expression for

Obviously, for each

By (

By successive substitutions in (

The series on the right in (

It can be easily verified that

So we can get

Since

Consider the following boundary value problem (BVP) with impulses:

Let us list some marks and conditions for convenience.

The nonlinearity

(H3) There exist

(H4) There exist

Then, we can get the following theorem.

Assume (H1), (H2), (H3), and (H4) are satisfied. And

First of all, we show that operator

Then

This implies that

It is easy to see that

We now proceed with the construction of the open sets

First, let

Let

Next, let

For

It follows from (i) of Theorem

This completes the proof.

Next, with

(H5) There exist

(H6) There exist

Assume (H1), (H2), (H5), and (H6) are satisfied. And

Let

First, let

Next, let

Then for

Then, we have

We see the case (ii) of Theorem

This completes the proof.

This work was supported by the NNSF of China under Grant no. 11271261, Natural Science Foundation of Shanghai (no. 12ZR1421600), Shanghai Municipal Education Commission (no. 10YZ74), the Slovenian Research Agency, and a Marie Curie International Research Staff Exchange Scheme Fellowship within the 7th European Community Programme (FP7-PEOPLE-2012-IRSES-316338).