- Research Article
- Open Access

# Positive Solutions to Nonlinear First-Order Nonlocal BVPs with Parameter on Time Scales

- Chenghua Gao
^{1}Email author and - Hua Luo
^{2}

**Received:**4 May 2010**Accepted:**3 June 2010**Published:**27 June 2011

## Abstract

We discuss the existence of solutions for the first-order multipoint BVPs on time scale : , , , where is a parameter, is a fixed number, , is continuous, is regressive and rd-continuous, , , , , and . For suitable , some existence, multiplicity, and nonexistence criteria of positive solutions are established by using well-known results from the fixed-point index.

## Keywords

- Positive Constant
- Real Line
- Closed Subset
- Main Tool
- Index Theory

## 1. Introduction

where is a fixed number, , is continuous, is regressive and rd-continuous, , and , is defined in its standard form; see [1, page 59] for details.

The multipoint boundary value problems arise in a variety of different areas of applied mathematics and physics. For example, the vibrations of a guy wire of a uniform cross-section and composed of parts of different densities can be set up as a multipoint boundary value problem [2]; also many problems in the theory of elastic stability can be handled by a multipoint problem [3]. So, the existence of solutions to multipoint boundary value problems have been studied by many authors; see [4–13] and the reference therein. Especially, in recent years the existence of positive solutions to multipoint boundary value problems on time scales has caught considerable attention; see [10–14]. For other background on dynamic equations on time scales, one can see [1, 15–18].

where is continuous, is regressive and rd-continuous, and . The existence results are based on Krasnoselskii's fixed-point theorem in cones and Leggett-Williams's theorem.

As we can see, if we take , , , and for , then (1.1) is reduced to (1.2). Because of the influence of the parameter , it will be more difficult to solve (1.1) than to solve (1.2).

For suitable , they gave some existence, multiplicity, and nonexistence criteria of positive solutions.

Motivated by the above results, by using the well-known fixed-point index theory [16, 19], we attempt to obtain some existence, multiplicity and nonexistence criteria of positive solutions to (1.1) for suitable .

The rest of this paper is arranged as follows. Some preliminary results including Green's function are given in Section 2. In Section 3, we obtain some useful lemmas for the proof of the main result. In Section 4, some existence and multiplicity results are established. At last, some nonexistence results are given in Section 5.

## 2. Preliminaries

Throughout the rest of this paper, we make the following assumptions:

is continuous and for ,

is rd-continuous, which implies that (where is defined in [16, 18, 20]).

Our main tool is the well-known results from the fixed-point index, which we state here for the convenience of the reader.

Theorem 2.1 (see [19]).

Let be a Banach space and be a cone in . For , we define . Assume that is completely continuous such for .

Let be equipped with the norm . It is easy to see that is a Banach space.

Lemma 2.2.

Proof.

Lemma 2.3.

Proof.

When , ,

When ,

Lemma 2.4.

Green's function has the following properties.

(i) ,

(ii) where

(iii) ,

Proof.

This proof is similar to [13, Lemma ], so we omit it.

where . For , let and .

Similar to the proof of [13, Lemma ], we can see that is completely continuous. By the above discussions, its not difficult to see that being a solution of BVP (1.1) equals the solution that is a fixed point of the operator .

## 3. Some Lemmas

## 4. Some Existence and Multiplicity Results

Theorem 4.1.

Proof.

This shows that has a fixed point in , which is a positive solution of the BVP (1.1).

This shows that has a fixed point in , which is another positive solution of the BVP (1.1).

Similar to the proof of Theorem 4.1, we have the following results.

Theorem 4.2.

Then,

(i)equation (1.1) has at least one positive solution if ,

(ii)equation (1.1) has at least one positive solution if ,

(iii)equation (1.1) has at least two positive solutions if .

Theorem 4.3.

Proof.

This shows that has a fixed point in , which is a positive solution of the BVP (1.1).

This shows that has a fixed point in , which is another positive solution of the BVP (1.1).

Similar to the proof of Theorem 4.3, we have the following results.

Theorem 4.4.

Then,

(i)equation (1.1) has at least one positive solution if ,

- (iii)
equation (1.1) has at least two positive solutions if .

Theorem 4.5.

Proof.

We only deal with the case that , . The other three cases can be discussed similarly.

which implies that the BVP (1.1) has at least one positive solution in .

Remark 4.6.

By making some minor modifications to the proof of Theorem 4.5, we can obtain the existence of at least one positive solution, if one of the following conditions is satisfied:

(i) , and .

(ii) , and .

(iii) , and .

(iv) , and .

Remark 4.7.

This is the condition of Theorem of [13]. So, our conclusions extend and improve the results of [13].

## 5. Some Nonexistence Results

Theorem 5.1.

Assume that (H1) and (H2) hold. If and , then the BVP (1.1) has no positive solutions for sufficiently small .

Proof.

We assert that the BVP (1.1) has no positive solutions for .

which is a contradiction.

Theorem 5.2.

Assume that (H1) and (H2) hold. If and , then the BVP (1.1) has no positive solutions for sufficiently large .

Proof.

We assert that the BVP (1.1) has no positive solutions for .

which is a contradiction.

Corollary 5.3.

Assume that (H1) and (H2) hold. If and , then the BVP (1.1) has no positive solutions for sufficiently large .

## Declarations

### Acknowledgments

This work was supported by the NSFC Young Item (no. 70901016), HSSF of Ministry of Education of China (no. 09YJA790028), Program for Innovative Research Team of Liaoning Educational Committee (no. 2008T054), the NSF of Liaoning Province (no. L09DJY065), and NWNU-LKQN-09-3

## Authors’ Affiliations

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