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

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

**2010**:235296

https://doi.org/10.1155/2010/235296

© Y. Lin and M. Pei. 2010

**Received: **4 December 2009

**Accepted: **29 March 2010

**Published: **10 May 2010

## Abstract

## 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);

Lemma 2.1.

Proof.

Lemma 2.2.

Proof.

Lemma 2.3 (see [20]).

## 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.

Proof.

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.

- (ii)

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.

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

## References

- Agarwal RP, Bohner M:
**Basic calculus on time scales and some of its applications.***Results in Mathematics. Resultate der Mathematik*1999,**35**(1-2):3-22.MATHMathSciNetView ArticleGoogle Scholar - Bohner M, Peterson A:
*Dynamic Equations on Time Scales: An Introduction with Application*. Birkhäuser, Boston, Mass, USA; 2001:x+358.View ArticleGoogle Scholar - Bohner M, Peterson A:
*Advances in Dynamic Equations on Time Scales*. Birkhäuser, Boston, Mass, USA; 2003:xii+348.MATHView ArticleGoogle Scholar - Agarwal RP, Hong H-L, Yeh C-C:
**The existence of positive solutions for the Sturm-Liouville boundary value problems.***Computers & Mathematics with Applications*1998,**35**(9):89-96. 10.1016/S0898-1221(98)00060-1MATHMathSciNetView ArticleGoogle Scholar - Erbe L, Peterson A:
**Positive solutions for a nonlinear differential equation on a measure chain.***Mathematical and Computer Modelling*2000,**32**(5-6):571-585. 10.1016/S0895-7177(00)00154-0MATHMathSciNetView ArticleGoogle Scholar - Hao Z-C, Xiao T-J, Liang J:
**Existence of positive solutions for singular boundary value problem on time scales.***Journal of Mathematical Analysis and Applications*2007,**325**(1):517-528. 10.1016/j.jmaa.2006.01.083MATHMathSciNetView ArticleGoogle Scholar - Agarwal RP, O'Regan D:
**Nonlinear boundary value problems on time scales.***Nonlinear Analysis: Theory, Methods & Applications*2001,**44**(4):527-535. 10.1016/S0362-546X(99)00290-4MATHMathSciNetView ArticleGoogle Scholar - Chyan CJ, Wong PJY:
**Triple solutions of focal boundary value problems on time scale.***Computers & Mathematics with Applications*2005,**49**(7-8):963-979. 10.1016/j.camwa.2005.01.015MATHMathSciNetView ArticleGoogle Scholar - Sun J-P, Li W-T:
**Existence of positive solutions to semipositone Dirichlet BVPs on time scales.***Dynamic Systems and Applications*2007,**16**(3):571-578.MATHMathSciNetGoogle Scholar - Sun J-P, Li W-T:
**Solutions and positive solutions to semipositone Dirichlet BVPs on time scales.***Dynamic Systems and Applications*2008,**17**(2):303-311.MATHMathSciNetGoogle Scholar - Agarwal RP, Wong F-H:
**Existence of positive solutions for non-positive higher-order BVPs.***Journal of Computational and Applied Mathematics*1998,**88**(1):3-14. 10.1016/S0377-0427(97)00211-2MATHMathSciNetView ArticleGoogle Scholar - Wong PJY, Agarwal RP:
**On eigenvalue intervals and twin eigenfunctions of higher-order boundary value problems.***Journal of Computational and Applied Mathematics*1998,**88**(1):15-43. 10.1016/S0377-0427(97)00202-1MATHMathSciNetView ArticleGoogle Scholar - Wong F-H:
**An application of Schauder's fixed point theorem with respect to higher order BVPs.***Proceedings of the American Mathematical Society*1998,**126**(8):2389-2397. 10.1090/S0002-9939-98-04709-1MATHMathSciNetView ArticleGoogle Scholar - Wong PJY, Agarwal RP:
**On the existence of solutions of singular boundary value problems for higher order difference equations.***Nonlinear Analysis: Theory, Methods &Applications*1997,**28**(2):277-287. 10.1016/0362-546X(95)00151-KMATHMathSciNetView ArticleGoogle Scholar - Liu J, Sang Y:
**Existence results for higher-order boundary value problems on time scales.***Advances in Difference Equations*2009,**2009:**-18.Google Scholar - Grossinho MR, Minhós FM:
**Upper and lower solutions for higher order boundary value problems.***Nonlinear Studies*2005,**12**(2):165-176.MATHMathSciNetGoogle Scholar - Agarwal RP:
*Boundary Value Problems for Higher Order Differential Equations*. World Scientific, Singapore; 1986:xii+307.MATHView ArticleGoogle Scholar - Lin Y, Pei M:
**Positive solutions for two-point semipositone right focal eigenvalue problem.***Boundary Value Problems*2007,**2007:**-12.Google Scholar - Lin Y, Pei M:
**Positive solutions of two-point right focal eigenvalue problems on time scales.***Advances in Difference Equations*2007,**2007:**-15.Google Scholar - Guo D, Lakshmikantham V:
*Nonlinear Problems in Abstract Cones, Notes and Reports in Mathematics in Science and Engineering*.*Volume 5*. Academic Press, Boston, Mass, USA; 1988:viii+275.Google Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.