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This paper makes a study on the existence of positive solution to

Initiated by Hilger in his Ph.D. thesis [

Recent research results indicate that considerable achievement has been made in the existence problems of positive solutions to dynamic equations on time scales. For details, please see [

Throughout this paper, we denote the

For convenience, we think of the blanket as an assumption that

We would like to mention the results of Sun and Li [

It is also noted that the researchers mentioned above [

Let

The rest of the paper is organized as follows. In Section

For convenience, we now give some symmetric definitions.

The interval

We note that such a symmetric time scale

A function

We say

Basic definitions on time scale can be found in [

Throughout this paper, it is assumed that

Let

Assume that

Assume that

From the previous lemma we know that

The operator

In addition, it is easy to see that the operator

Now, we provide some background material from the theory of cones in Banach spaces [

Firstly, we list the Krasnosel'skii's fixed point theorem [

Let

Given a nonnegative continuous functional

Secondly, we state the generalized Avery-Henderson fixed point theorem [

Let

The following lemma can be found in [

Let

Let

Finally, we list the fixed point theorem due to Avery-Peterson [

Let

Let

In this subsection, we discuss the existence of single positive symmetric solution of the problem (

Problem (

We divide the proof into two cases.

In view of

If

From

Since

Now, we consider

Suppose that

Suppose that

In this subsection, we discuss the existence of positive symmetric solutions to problems (

First, we will state and prove the following main result of problem (

Suppose that the following conditions hold:

there exists constant

there exists constant

Without loss of generality, we may assume that

Let

Now, set

In this subsection, under the conditions

Suppose that

It is easy to see that under the assumptions, conditions (i) and (ii) in Theorem

Suppose that

Firstly, let

Nextly, for

Assume that

Assume that

From

Consequently, Theorem

From the proof of Theorems

Suppose that

Suppose that

Suppose that

First, in view of

Suppose that

On one hand, since

From Theorems

Assume that

Assume that

In this subsection, under

Combining the proofs of Theorems

Suppose that

Suppose that

In the previous section, we have obtained some results on the existence of at least single or twin positive symmetric solutions to problem (

Based on the obtained symmetric solution position and local properties, we can only get some local properties of solutions by using method one; however, the position of solutions is not determined. In contrast, by means of method two, we cannot only get some local properties of solutions but also give the position of all solutions, with regard to some subsets of the cone, which has to meet some conditions which are stronger than those of method one. Obviously, the local properties of obtained solutions are different by using the two different methods. Hence, it is convenient for us to comprehensively comprehend the solutions of the models by using the two different techniques.

In Section

For the notational convenience, we denote

In this subsection, in view of the generalized Avery-Henderson fixed-point theorem [

For

We now present the results in this subsection.

If there are positive numbers

By the definition of completely continuous operator

Firstly, we verify that if

If

Secondly, we show that

If we choose

Finally, we prove that

In fact, the constant function

From Theorem

Let

When

Using Lemma

Suppose that there are positive numbers

From Theorem

Let

Assume that

there exists

First, by condition (ii), let

Second, choose

Third, choose

From above analysis, we get

In terms of Theorem

Assume that

there exists

In this subsection, the existence criteria for at least

Define the nonnegative continuous convex functionals

Now, we list and prove the results in this subsection.

Suppose that there exist constants

By the definition of completely continuous operator

For all

Firstly, we show that

For any

Secondly, we verify that condition (i) of Lemma

Thirdly, we prove that condition (ii) of Lemma

Finally, we check condition (iii) of Lemma

We remark that condition (i) in Theorem

If condition (i) in Theorem

By Theorem

Suppose on the contrary that for any

Let

Let

Similar to the proof of Theorem

In this section, we give two simple examples to illustrate that the conclusions we will arrive at are different with their own distinctive advantages.

Let

Consider the following boundary value problem with

Let

Consider the following boundary value problem:

It is obvious that (i), (ii), and (iii) in Theorem

However, for arbitrary positive numbers

This work was supported by the Grant of Department of Education Jiangsu Province (09KJD110006) and Science Foundation of Lanzhou University of Technology (BS10200903).