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

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## Abstract

We discuss the existence of solutions for the first-order multipoint BVPs on time scale Open image in new window : Open image in new window , Open image in new window , Open image in new window , where Open image in new window is a parameter, Open image in new window is a fixed number, Open image in new window , Open image in new window is continuous, Open image in new window is regressive and rd-continuous, Open image in new window , Open image in new window , Open image in new window , Open image in new window , and Open image in new window . For suitable Open image in new window , 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 Open image in new window is a fixed number, Open image in new window , Open image in new window is continuous, Open image in new window is regressive and rd-continuous, Open image in new window , and Open image in new window , Open image in new window 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 Open image in new window 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, 5, 6, 7, 8, 9, 10, 11, 12, 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, 11, 12, 13, 14]. For other background on dynamic equations on time scales, one can see [1, 15, 16, 17, 18].

where Open image in new window is continuous, Open image in new window is regressive and rd-continuous, Open image in new window and Open image in new window . 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 Open image in new window , Open image in new window , Open image in new window , and Open image in new window for Open image in new window , then (1.1) is reduced to (1.2). Because of the influence of the parameter Open image in new window , it will be more difficult to solve (1.1) than to solve (1.2).

For suitable Open image in new window , 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 Open image in new window .

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:

Open image in new window is continuous and Open image in new window for Open image in new window ,

Open image in new window is rd-continuous, which implies that Open image in new window (where Open image in new window 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 Open image in new window be a Banach space and Open image in new window be a cone in Open image in new window . For Open image in new window , we define Open image in new window . Assume that Open image in new window is completely continuous such Open image in new window for Open image in new window .

Let Open image in new window be equipped with the norm Open image in new window . It is easy to see that Open image in new window is a Banach space.

Lemma 2.2.

Proof.

Lemma 2.3.

Proof.

When Open image in new window , Open image in new window ,

When Open image in new window ,

Lemma 2.4.

Green's function Open image in new window has the following properties.

(i) Open image in new window ,

(ii) Open image in new window where Open image in new window Open image in new window

(iii) Open image in new window , Open image in new window

Proof.

This proof is similar to [13, Lemma Open image in new window ], so we omit it.

where Open image in new window . For Open image in new window , let Open image in new window and Open image in new window .

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

## 3. Some Lemmas

Lemma 3.1.

Proof.

Lemma 3.2.

Proof.

Lemma 3.3.

where Open image in new window ; Open image in new window .

Proof.

## 4. Some Existence and Multiplicity Results

Theorem 4.1.

Proof.

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

This shows that Open image in new window has a fixed point in Open image in new window , 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 Open image in new window ,

(ii)equation (1.1) has at least one positive solution if Open image in new window ,

(iii)equation (1.1) has at least two positive solutions if Open image in new window .

Theorem 4.3.

Proof.

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

This shows that Open image in new window has a fixed point in Open image in new window , 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 Open image in new window ,

- (iii)
equation (1.1) has at least two positive solutions if Open image in new window .

Theorem 4.5.

Proof.

We only deal with the case that Open image in new window , Open image in new window . The other three cases can be discussed similarly.

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

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) Open image in new window , Open image in new window and Open image in new window .

(ii) Open image in new window , Open image in new window and Open image in new window .

(iii) Open image in new window , Open image in new window and Open image in new window .

(iv) Open image in new window , Open image in new window and Open image in new window .

Remark 4.7.

This is the condition of Theorem Open image in new window 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 Open image in new window and Open image in new window , then the BVP (1.1) has no positive solutions for sufficiently small Open image in new window .

Proof.

We assert that the BVP (1.1) has no positive solutions for Open image in new window .

which is a contradiction.

Theorem 5.2.

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

Proof.

We assert that the BVP (1.1) has no positive solutions for Open image in new window .

which is a contradiction.

Corollary 5.3.

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

## Notes

### 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

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