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We study a general second-order

The theory of dynamic equations on time scales unifies the well-known analogies in the concept of difference equations and differential equations. In the past few years, the boundary value problems of dynamic equations on time scales have been studied by many authors (see [

In 2008, Lin and Du [

In 2009, Topal and Yantir [

Impulsive differential equations are now recognized as an excellent source of models for simulating processes and phenomena observed in control theory, physics, chemistry, population dynamics, biotechnology, industrial robotics, optimal control, and so forth. In recent years, impulsive differential equations have become a very active area of research. In this paper, we consider the following impulsive singular dynamic equations on time scales:

The main theorems of this paper complement the very little existence results devoted to impulsive dynamic equations on a time scale. We will prove our two existence results for the problem (

In this section we state the preliminary information that we need to prove the main results. From Lemmas 2.1 and 2.2 in [

Assume that (C3) holds. Then the equations

(a)

For the rest of the paper we need the following assumption:

Assume that (C3) and (C4) hold. Let

Green’s function

Assume that (C3) and (C4) hold. Let

Green’s function

Assume that (C1)–(C4) hold. Then the solution of (

The proofs of the Lemmas

For our constructions, we will consider the Banach space

Thus, from Lemma

The following content will play major role in our next analysis.

Let

Assume

Suppose that

Let

First, by using Theorem

Suppose that conditions

for

There exists

for any

Consider that

Since (

Let

From (

Similarly, from (

Thus, for any

Next, for any

So the conditions of Theorems

Consider the following singular boundary value problem:

Let

In the next, using Theorem

Suppose that conditions

First we will show that there exists a solution

To see this, let

Next we show

We have

Now Theorem

Similarly, if we put

Similar to the proof of Theorem

Suppose that conditions

If in (

It is easy to use Theorem

Suppose that conditions (C1)–(C4); (

Consider the boundary value problem

Then (

To see this we will apply Theorem

The authors declare that they have no conflict of interests regarding the publication of this paper.

The work was supported by Youth academic backbone project of Heilongjiang Provincial University (no. 1252G035).