^{1}

^{2}

^{2}

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^{1}

^{2}

^{3}

By using the well-known Schauder fixed point theorem and upper and lower solution method, we present some existence criteria for positive solution of an

Initiated by Hilger in his Ph.D. thesis [

Here and hereafter, we denote

Recent research results indicate that considerable work has been made in the existence problems of solutions of boundary value problems on time scales, for details, see [

Motivated by references [

For the convenience of statements, now we present some basic definitions and lemmas concerning the calculus on time scales that one needs to read this manuscript, which can be found in [

A time scale

We say that a function

If

If

A function

Next, we list some lemmas which will be used in the sequel.

Suppose

If

If

If

If

Suppose

Let

Suppose

Now, we can obtain the following lemma which is similar to Lemma

Suppose

Throughout this paper, it is assumed that

Define the Banach space

To demonstrate existence of positive solutions to problem (

Now we state and prove our main result.

Let

For each

There exists a function

There exists a function

It follows from the condition (A1) that

Consider the

Suppose

We first show that

Assume that

If

If

However, by (

Assume that

Assume that

Assume that

(a) If

(b) If

Assume that

Assume that there exist sequences

(c) If there exist sequences

Thus, Cases

Essentially the same reasoning as the proof of inequality (

Hence

Now, we discuss the boundary value problem

Essentially the same reasoning as the proof of inequality (

If there exists

By using the similar arguments as above, we have

So there exists a positive number

Now, we define a function

If we replace

If we replace

If we replace

So it is easily obtain the analogue of Theorem

If (A3) is appropriately adjusted, we can replace

Assume that (H1)–(H3), (A1) and (A2) hold, and in addition suppose the following conditions are satisfied:

(A4)

(A5) There exists a function

(A6)

Then the result in Theorem

Let

In this section, we consider how to construct a lower solution

Assume that there exists a nonincreasing positive sequence

Let

Without loss of generality,

Now we discuss how to construct a lower solution

(A7) For each

(A8) There exists a function

Let

By Corollary

From Lemma

Assume

We can replace

Looking at Theorem

Let

Denote

From Theorems

Let

Without loss of generality suppose

In this section, we present an example to illustrate our results. Let

Now we show that (A8) holds with

Notice that if

If

If

This paper is supported by XZIT under Grant XKY2008311 and DEGP under Grant 0709-03.