Open Access

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

Boundary Value Problems20092009:685040

DOI: 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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq1_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq2_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq3_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq4_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq5_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq6_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq7_HTML.gif . 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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq8_HTML.gif be a time scale, that is, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq9_HTML.gif is an arbitrary nonempty closed subset of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq10_HTML.gif . For each interval https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq11_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq12_HTML.gif we define https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq13_HTML.gif 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:

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_Equ1_HTML.gif
(11)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq14_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq15_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq16_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq17_HTML.gif . 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:

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_Equ2_HTML.gif
(12)

Throughout this paper, we always assume that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq18_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq19_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq20_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq21_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq22_HTML.gif

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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq23_HTML.gif , we impose the following hypotheses:

(H1) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq24_HTML.gif is continuous;

(H2)for fixed https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq25_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq26_HTML.gif is monotone increasing on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq27_HTML.gif ;

(H3)there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq28_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_Equ3_HTML.gif
(13)

Remark 1.1.

If (H3) is satisfied, then
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_Equ4_HTML.gif
(14)

2. Main Results

Lemma 2.1.

The BVP (1.2) is equivalent to the integral equation
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_Equ5_HTML.gif
(21)
where
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_Equ6_HTML.gif
(22)
is called the Green's function for the corresponding linear BVP, here
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_Equ7_HTML.gif
(23)
is the Green's function for the BVP:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_Equ8_HTML.gif
(24)

Proof.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq29_HTML.gif be a solution of the BVP:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_Equ9_HTML.gif
(25)
Then, it is easy to know that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_Equ10_HTML.gif
(26)
Now, if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq30_HTML.gif is a solution of the BVP (1.2), then it can be expressed by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_Equ11_HTML.gif
(27)
which together with the boundary conditions in (1.2) and (2.6) implies that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_Equ12_HTML.gif
(28)

On the other hand, if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq31_HTML.gif satisfies (2.1), then it is easy to verify that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq32_HTML.gif is a solution of the BVP (1.2).

Lemma 2.2.

For any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq33_HTML.gif one has
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_Equ13_HTML.gif
(29)

Proof.

Since it is obvious from the expression of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq34_HTML.gif that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_Equ14_HTML.gif
(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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq35_HTML.gif . Furthermore, there exist https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq36_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_Equ15_HTML.gif
(211)

Proof.

Let
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_Equ16_HTML.gif
(212)
Define an operator https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq37_HTML.gif :
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_Equ17_HTML.gif
(213)

Then it is obvious that fixed points of the operator https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq38_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq39_HTML.gif are positive solutions of the BVP (1.2).

First, in view of (H2), it is easy to know that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq40_HTML.gif is increasing.

Next, we may assert that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq41_HTML.gif , which implies that for any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq42_HTML.gif , there exist positive constants https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq43_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq44_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_Equ18_HTML.gif
(214)
In fact, for any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq45_HTML.gif , there exist https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq46_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_Equ19_HTML.gif
(215)
which together with (H2), (H3), and Remark 1.1 implies that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_Equ20_HTML.gif
(216)
By Lemma 2.2 and (2.16), for any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq47_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_Equ21_HTML.gif
(217)
If we let
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_Equ22_HTML.gif
(218)
then it follows from (2.17) and (2.18) that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_Equ23_HTML.gif
(219)

which shows that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq48_HTML.gif .

Now, for any fixed https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq49_HTML.gif , we denote
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_Equ24_HTML.gif
(220)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_Equ25_HTML.gif
(221)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_Equ26_HTML.gif
(222)
and let
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_Equ27_HTML.gif
(223)
where
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_Equ28_HTML.gif
(224)
Then, it is easy to know from (2.20), (2.21), (2.22), (2.23), (2.24), (H3), and Remark 1.1 that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_Equ29_HTML.gif
(225)
Moreover, if we let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq50_HTML.gif , then it follows from (2.22), (2.23), (2.24), and (H3) by induction that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_Equ30_HTML.gif
(226)
which together with (2.25) implies that for any positive integers https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq51_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq52_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_Equ31_HTML.gif
(227)
Therefore, there exists a https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq53_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq54_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq55_HTML.gif converge uniformly to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq56_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq57_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_Equ32_HTML.gif
(228)
Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq58_HTML.gif is increasing, in view of (2.28), we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_Equ33_HTML.gif
(229)
So,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_Equ34_HTML.gif
(230)
which shows that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq59_HTML.gif is a positive solution of the BVP (1.2). Furthermore, since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq60_HTML.gif , there exist https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq61_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_Equ35_HTML.gif
(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 Article
  2. Bohner M, Peterson A: Dynamic Equations on Time Scales: An Introduction with Application. Birkhäuser, Boston, Mass, USA; 2001:x+358.View Article
  3. Bohner M, Peterson A: Advances in Dynamic Equations on Time Scales. Birkhäuser, Boston, Mass, USA; 2003:xii+348.MATHView Article
  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 Article
  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 Article
  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 Article
  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.MathSciNet
  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 Article
  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.MathSciNet
  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 Article

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.