Open Access

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

Boundary Value Problems20092009:685040

https://doi.org/10.1155/2009/685040

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 [15]. 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:

(11)

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 [810] 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:

(12)

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 ;

(H3)there exists such that
(13)

Remark 1.1.

If (H3) is satisfied, then
(14)

2. Main Results

Lemma 2.1.

The BVP (1.2) is equivalent to the integral equation
(21)
where
(22)
is called the Green's function for the corresponding linear BVP, here
(23)
is the Green's function for the BVP:
(24)

Proof.

Let be a solution of the BVP:
(25)
Then, it is easy to know that
(26)
Now, if is a solution of the BVP (1.2), then it can be expressed by
(27)
which together with the boundary conditions in (1.2) and (2.6) implies that
(28)

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.

For any one has
(29)

Proof.

Since it is obvious from the expression of that
(210)

we know that (2.9) is fulfilled.

Our main result is the following theorem.

Theorem 2.3.

Assume that (H1)–(H3) are satisfied. Then, the BVP (1.2) has at least one positive solution . Furthermore, there exist such that
(211)

Proof.

Let
(212)
Define an operator :
(213)

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.

Next, we may assert that , which implies that for any , there exist positive constants and such that
(214)
In fact, for any , there exist such that
(215)
which together with (H2), (H3), and Remark 1.1 implies that
(216)
By Lemma 2.2 and (2.16), for any , we have
(217)
If we let
(218)
then it follows from (2.17) and (2.18) that
(219)

which shows that .

Now, for any fixed , we denote
(220)
(221)
(222)
and let
(223)
where
(224)
Then, it is easy to know from (2.20), (2.21), (2.22), (2.23), (2.24), (H3), and Remark 1.1 that
(225)
Moreover, if we let , then it follows from (2.22), (2.23), (2.24), and (H3) by induction that
(226)
which together with (2.25) implies that for any positive integers and ,
(227)
Therefore, there exists a such that and converge uniformly to on and
(228)
Since is increasing, in view of (2.28), we have
(229)
So,
(230)
which shows that is a positive solution of the BVP (1.2). Furthermore, since , there exist such that
(231)

Declarations

Acknowledgment

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

Authors’ Affiliations

(1)
Department of Applied Mathematics, Lanzhou University of Technology

References

  1. 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
  2. Bohner M, Peterson A: Dynamic Equations on Time Scales: An Introduction with Application. Birkhäuser, Boston, Mass, USA; 2001:x+358.View ArticleGoogle Scholar
  3. Bohner M, Peterson A: Advances in Dynamic Equations on Time Scales. Birkhäuser, Boston, Mass, USA; 2003:xii+348.MATHView ArticleGoogle Scholar
  4. 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
  5. 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
  6. 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
  7. 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
  8. 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
  9. 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
  10. 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

© Jian-Ping Sun 2009

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.