The study of dynamic equations on time scales goes back to its founder Hilger [3] and it is a new area of still fairly theoretical exploration in mathematics. In the recent years boundary value problem on time scales has received considerable attention [4–6]. And an increasing interest in studying the existence of solutions to dynamic equations on time scales is observed, for example, see [7–16].

For convenience, we first recall some definitions and calculus on time scales, so that the paper is self-contained. For the further details concerning the time scales, please see [17–19] which are excellent works for the calculus of time scales.

A time scale
is an arbitrary nonempty closed subset of real numbers
. The operators
and
from
to

are called the forward jump operator and the backward jump operator, respectively.

For all
we assume throughout that
has the topology that it inherits from the standard topology on
The notations
and so on, will denote time-scale intervals

where
with

Definition 2.1.

Fix

Let

Then we define

to be the number (if it exists) with the property that given

there is a neighborhood

of

with

Then
is called derivative of

Definition 2.2.

If

then we define the integral by

We say that a function

is regressive provided

where

which is called graininess function. If

is a regressive function, then the generalized exponential function

is defined by

for

is the cylinder transformation, which is defined by

Let

be two regressive functions, then define

The generalized function
has then the following properties.

Lemma 2.3 (see [18]).

Assume that
are two regressive functions, then

(i)
and

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

The following well-known fixed point theorem will play a very important role in proving our main result.

Theorem 2.4 (see [20]).

Let

be a Banach space, and let

be completely continuous. Assume that

is a bounded linear operator such that

is not an eigenvalue of

and

Then
has a fixed point in

Throughout this paper, let

be endowed with the norm by

where
And we make the following assumptions:

()
and

()
and

()

Set

For convenience, we denote

First, we present two lemmas about the calculus on Green functions which are crucial in our main results.

Lemma 2.5.

Assume that

and

are satisfied. If

then

is a solution of the following boundary value problem (BVP):

where the Green's function of (2.13) is as follows:

where
are given as (2.12), respectively.

Proof.

If

is a solution of (2.13), setting

then it follows from the first equation of (2.13) that

Multiplying (2.17) by

and integrating from

to

we get

Similarly, by (2.18), we have

Then substituting (2.18) into (2.19), we get for each

that

Substituting this expression for

into the boundary conditions of (2.13). By some calculations, we get

Then substituting (2.21) into (2.20), we get

By interchanging the order of integration and some rearrangement of (2.22), we obtain

Thus, we obtain (2.14) consequently.

On the other hand, if

satisfies (2.14), then direct differentiation of (2.14) yields

And it is easy to know that
and
satisfies (2.13).

Corollary 2.6.

If

then BVP (2.13) reduces to the following problem:

From Lemma 2.5, BVP (2.25) has a unique solution

where the Green's function of (2.25) is as follows:

Proof.

If

is a solution of (2.25), take

then

Hence, from (2.20) we have

Substituting this expression for

into the boundary conditions of (2.25). By some calculations, we obtain

where

is given as (2.28). Then substituting (2.31) into (2.30), we get

where
are as in (2.29), respectively. By some rearrangement of (2.32), we obtain (2.26) consequently.

From the proof of Corollary 2.6, if
take
we get the following result.

Corollary 2.7.

The following boundary value problem:

where the Green's function of (2.33) is as follows:

After some rearrangement of (2.35), one obtains

Remark 2.8.

Green function (2.37) associated with BVP (2.33) which is a special case of (2.13) is coincident with part of [21, Lemma 1].

Lemma 2.9.

Assume that conditions (

)–(

) are satisfied. If

then boundary value problem

and
is given in Lemma 2.5.

Proof.

Consider the following boundary value problem:

The Green's function associated with the BVP (2.41) is
. This completes the proof.

Remark 2.10.

In [

1, Lemma 2.5], the solution of (1.2) is defined as

where
and
are given as (1.4) and (1.5), respectively. In fact,
is incorrect. Thus, we give the right form of
as the special case
in our Lemma 2.9.