# Existence of Positive Solutions for Semipositone Higher-Order BVPS on Time Scales

- Yuguo Lin
^{1}and - Minghe Pei
^{1}Email author

**2010**:235296

**DOI: **10.1155/2010/235296

© Y. Lin and M. Pei. 2010

**Received: **4 December 2009

**Accepted: **29 March 2010

**Published: **10 May 2010

## Abstract

We offer conditions on semipositone function such that the boundary value problem, , , , , , , , has at least one positive solution, where is a time scale and is continuous with for some positive constant .

## 1. Introduction

Throughout this paper, let be a time scale, for any , the interval defined as . Analogous notations for open and half-open intervals will also be used in the paper. We also use the notation to denote the real interval . To understand further knowledge about dynamic equations on time scales, the reader may refer to [1–3] for an introduction to the subject.

Throughout, we assume that is continuous with for some positive constant .

We say that is a positive solution of BVP (1.1) and (1.2), if is a solution of BVP (1.1) and (1.2) and .

Various cases of BVP (1.1) and (1.2) have attracted a lot of attention in the literature. When , BVP (1.1) and (1.2) has been studied by many specialists. For example, Agarwal et al. [4] have established the existence of positive solutions for continuous case of the semipositone Sturm-Liouville BVPs. Erbe and Peterson [5] andHao et al.[6]dealt with Sturm-Liouville BVPs on time scale of positone nonlinear term. In addition, Agarwal and O'Regan [7] obtained positive solution of second-order right focal BVPs on time scale by using nonlinear alternative of Leray-Schauder type. In 2005, Chyan and Wong [8] obtained triple solutions of the same BVPs with [7]. Recently,Sun and Li[9, 10] investigated semipositone Dirichlet BVPs on time scale. For higher-order BVPs, continuous case of BVP (1.1) and (1.2) have been investigated by Agarwal and Wong [11], Wong and Agarwal [12] and Wong [13]. The discrete positone case of BVP (1.1) and (1.2) has been tackled by using a fixed point theorem for mappings that are decreasing with respect to a cone in [14]. Especially, time-scale case of (1.1) with four-point boundary condition has been studied by Liu and Sang [15]. Besides, BVP (1.1) and (1.2) of nonlinear positone term which satisfied Nagumo-type conditions have been dealt with in [16]. Motivated by the works mentioned above, the purpose of this paper is to tackle semipositone BVP (1.1) and (1.2). In fact, BVPs appeared in [7–14] can be looked at as special case of BVP (1.1) and (1.2) in this paper. For other related works, we also refer to [17–19].

The paper is outlined as follows. In Section 2, we will present some notations and lemmas which will be used later. In Section 3, by using Krasnoselskii's fixed point theorem in a cone, we offer criteria for the existence of positive solution of BVP (1.1) and (1.2).

## 2. Preliminary

In this section, we offer some notations and lemmas, which will be used in main results. Throughout this paper, we always use the following notations:

(*C*_{1})
is the Green's function of the differential equation
subject to the boundary conditions (1.2);

*C*

_{2}) is the Green's function of the differential equation subject to the boundary conditions

*C*

_{3}) Define , as

Lemma 2.1.

Proof.

Lemma 2.2.

and is a positive constant.

Proof.

Lemma 2.3 (see [20]).

- (i)
or

- (ii)

Then, has a fixed point in .

## 3. Main Results

In this section, by using Lemma 2.3, we offer criteria for the existence of positive solution of BVP (1.1) and (1.2).

Lemma 3.1.

The operator maps into .

Proof.

Hence, maps into .

Lemma 3.2.

The operator is completely continuous.

Proof.

This shows that is continuous.

The Arzela-Ascoli theorem guarantees that is relatively compact, so is completely continuous.

Theorem 3.3.

- (i)there exist such that for any ,(3.22)

- (ii)with such that for any ,(3.23)where is given in (3.4), are given in (3.5), are given in (3.7), and(3.24)

Then BVP (1.1) and (1.2) has a positive solution.

Proof.

where is given in (3.7) and is given in (3.24).

Therefore, it follows from Lemma 2.3 that BVP (3.8) has a solution such that .

So, is a positive solution of BVP (1.1) and (1.2). This completes the proof.

Corollary 3.4.

- (a)for any ,(3.36)
where is a continuous function which is nondecreasing in for each fixed and is a continuous nonnegative function on ,

- (b)for any ,(3.37)
where is a continuous function which is nondecreasing in for each fixed and is a continuous nonnegative function on ,

- (c)there exists such that(3.38)
- (d)there exists with such that(3.39)

Then BVP (1.1) and (1.2) has a positive solution.

Proof.

So, condition ii of Theorem 3.3 is satisfied.

Therefore, from Theorem 3.3, BVP(1.1) and (1.2) has a positive solution.

Corollary 3.5.

where . Then BVP (1.1) and (1.2) has one positive solution.

Proof.

So, condition of Theorem 3.3 is satisfied.

Finally we present an example to illustrate our result.

Example 3.6.

Hence, conditions and in Corollary 3.4 are satisfied. Therefore from Corollary 3.4, (3.46) has at least one positive solution.

Remark 3.7.

In Example 3.6, because nonlinear term may attain negative value, the result in [15] is not applicable.

## Declarations

### Acknowledgment

The authors thank the referee for valuable suggestions which led to improvement of the original manuscript.

## Authors’ Affiliations

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