# Existence of Positive Solution to Second-Order Three-Point BVPs on Time Scales

- Jian-Ping Sun
^{1}Email author

**2009**:685040

**DOI: **10.1155/2009/685040

© Jian-Ping Sun 2009

**Received: **19 April 2009

**Accepted: **14 September 2009

**Published: **28 September 2009

## Abstract

We are concerned with the following nonlinear second-order three-point boundary value problem on time scales , , , , where with and . A new representation of Green's function for the corresponding linear boundary value problem is obtained and some existence criteria of at least one positive solution for the above nonlinear boundary value problem are established by using the iterative method.

## 1. Introduction

Let be a time scale, that is, is an arbitrary nonempty closed subset of . For each interval of we define For more details on time scales, one can refer to [1–5]. Recently, three-point boundary value problems (BVPs for short) for second-order dynamic equations on time scales have received much attention. For example, in 2002, Anderson [6] studied the following second-order three-point BVP on time scales:

where , , and . Some existence results of at least one positive solution and of at least three positive solutions were established by using the well-known Krasnoselskii and Leggett-Williams fixed point theorems. In 2003, Kaufmann [7] applied the Krasnoselskii fixed point theorem to obtain the existence of multiple positive solutions to the BVP (1.1). For some other related results, one can refer to [8–10] and references therein.

In this paper, we are concerned with the existence of at least one positive solution for the following second-order three-point BVP on time scales:

Throughout this paper, we always assume that with , , and

It is interesting that the method used in this paper is completely different from that in [6, 7, 9, 10], that is, a new representation of Green's function for the corresponding linear BVP is obtained and some existence criteria of at least one positive solution to the BVP (1.2) are established by using the iterative method.

For the function , we impose the following hypotheses:

(H1) is continuous;

(H2)for fixed , is monotone increasing on ;

Remark 1.1.

## 2. Main Results

Lemma 2.1.

Proof.

On the other hand, if satisfies (2.1), then it is easy to verify that is a solution of the BVP (1.2).

Lemma 2.2.

Proof.

we know that (2.9) is fulfilled.

Our main result is the following theorem.

Theorem 2.3.

Proof.

Then it is obvious that fixed points of the operator in are positive solutions of the BVP (1.2).

First, in view of (H2), it is easy to know that is increasing.

which shows that .

## Declarations

### Acknowledgment

This work is supported by the National Natural Science Foundation of China (10801068).

## Authors’ Affiliations

## References

- Agarwal RP, Bohner M: Basic calculus on time scales and some of its applications.
*Results in Mathematics*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 - Hilger S: Analysis on measure chains—a unified approach to continuous and discrete calculus.
*Results in Mathematics*1990, 18(1–2):18–56.MATHMathSciNetView ArticleGoogle Scholar - Lakshmikantham V, Sivasundaram S, Kaymakcalan B:
*Dynamic Systems on Measure Chains, Mathematics and Its Applications*.*Volume 370*. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1996:x+285.View ArticleGoogle Scholar - Anderson DR: Solutions to second-order three-point problems on time scales.
*Journal of Difference Equations and Applications*2002, 8(8):673–688. 10.1080/1023619021000000717MATHMathSciNetView ArticleGoogle Scholar - Kaufmann ER: Positive solutions of a three-point boundary-value problem on a time scale.
*Electronic Journal of Differential Equations*2003, 2003(82):-11.MathSciNetGoogle Scholar - Khan RA, Nieto JJ, Otero-Espinar V: Existence and approximation of solution of three-point boundary value problems on time scales.
*Journal of Difference Equations and Applications*2008, 14(7):723–736. 10.1080/10236190701840906MATHMathSciNetView ArticleGoogle Scholar - Luo H, Ma Q: Positive solutions to a generalized second-order three-point boundary-value problem on time scales.
*Electronic Journal of Differential Equations*2005, 2005(17):-14.MathSciNetGoogle Scholar - Sun H-R, Li W-T: Positive solutions for nonlinear three-point boundary value problems on time scales.
*Journal of Mathematical Analysis and Applications*2004, 299(2):508–524. 10.1016/j.jmaa.2004.03.079MATHMathSciNetView ArticleGoogle 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.