For convenience, we list the following definitions which can be found in [1–5].

Definition 2.1.

A time scale is a nonempty closed subset of real numbers . For and , define the forward jump operator and backward jump operator , respectively, by

for all . If , is said to be right scattered, if , is said to be left scattered; if , is said to be right dense, and if , is said to be left dense. If has a right scattered minimum , define ; otherwise set . If has a left scattered maximum , define ; otherwise set .

Definition 2.2.

For and , the delta derivative of at the point is defined to be the number (provided that it exists), with the property that for each , there is a neighborhood of such that

for all .

For and , the nabla derivative of at , denoted by (provided it exists) with the property that for each , there is a neighborhood of such that

for all .

Definition 2.3.

A function is left-dense continuous (i.e., -continuous), if is continuous at each left-dense point in and its right-sided limit exists at each right-dense point in .

Definition 2.4.

If , then we define the delta integral by

If , then we define the nabla integral by

Lemma 2.5.

If condition holds, then for , the boundary value problem (BVP)

has the unique solution

Proof.

By caculating, we can easily get (2.7). So we omit it.

Lemma 2.6.

If condition holds, then for , the boundary value problem (BVP)

has the unique solution

where .

Proof.

Integrating both sides of equation in (2.8) on , we have

So,

By boundary value condition , we have

By (2.10) and (2.12) we know

This together with Lemma 2.5 implies that

where . The proof is complete.

Lemma 2.7.

Let condition holds If and , then the unique solution of (2.8) satisfies

Proof.

By , we can know that the graph of is concave down on , and is nonincreasing on . This together with the assumption that the boundary condition implies that for . This implies that

So we only prove By condition we have

The proof is completed.

Lemma 2.8.

Let condition hold. If and , then the unique positive solution of (BVP) (2.8) satisfies

where , .

Proof.

By , we can know that the graph of is concave down on , and is nonincreasing on . This together with the assumption that the boundary condition implies that for . This implies that

For all , we have from the concavity of that

that is,

This together with the boundary condition implies that

This completes the proof.

Let be endowed with the ordering if for all and is defined as usual by maximum norm. Clearly, it follows that is a Banach space.

For the convenience, let

We define two cones by

where , is defined in Lemma 2.8 and

Define the operators and by setting

where ,

where , , and . Obviously, is a solution of the BVP(1.6) if and only if is a fixed point of operator .

Lemma 2.9.

is completely continuous.

Proof.

It is easy to see that by and Lemma 2.8. By Arzela-Ascoli theorem and Lebesgue dominated convergence theorem, we can easily prove that operator is completely continuous.

Lemma 2.10 (see [20, 21]).

Let be a cone in a Banach space . Let be an open bounded subset of with and . Assume that is a compact map such that for . Then the following results hold.

(1)If , , then .

(2)If there exists such that for all and all , then .

(3)Let be open in such that . If and , then has a fixed point in . The same result holds if and , where denotes fixed point index.

We define

Lemma 2.11 (see [20]).

defined above has the following properties:

(a)

(b) is open relative to K;

(c) if and only if

(d)if , then for .

For the convenience, we introduce the following notations:

Remark 2.12.

By we can know that ,

Lemma 2.13.

If satisfies the following condition :

then

Proof.

For , then from (2.30) we have

So that

Therefore,

This implies that for . Hence by Lemma 2.10(1) it follows that .

Lemma 2.14.

If satisfies the following condition:

then

Proof.

Let for . Then . We claim that

In fact, if not, there exist and such that . By , we have

So that

For , then

This together with Lemma 2.11(c) implies that

a contradiction. Hence by Lemma 2.10(2) it follows that .