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

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

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

http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq18_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq19_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq20_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq21_HTML.gif , and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq23_HTML.gif , we impose the following hypotheses:

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

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

(H3)there exists http://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq28_HTML.gif such that
http://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
http://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
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_Equ5_HTML.gif
(21)
where
http://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
http://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:
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_Equ8_HTML.gif
(24)

Proof.

Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq29_HTML.gif be a solution of the BVP:
http://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
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_Equ10_HTML.gif
(26)
Now, if http://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
http://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
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_Equ12_HTML.gif
(28)

On the other hand, if http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq33_HTML.gif one has
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq34_HTML.gif that
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq35_HTML.gif . Furthermore, there exist http://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq36_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_Equ15_HTML.gif
(211)

Proof.

Let
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_Equ16_HTML.gif
(212)
Define an operator http://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq37_HTML.gif :
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq38_HTML.gif in http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq41_HTML.gif , which implies that for any http://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq42_HTML.gif , there exist positive constants http://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq43_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq44_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_Equ18_HTML.gif
(214)
In fact, for any http://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq45_HTML.gif , there exist http://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq46_HTML.gif such that
http://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
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq47_HTML.gif , we have
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_Equ21_HTML.gif
(217)
If we let
http://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
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_Equ23_HTML.gif
(219)

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

Now, for any fixed http://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq49_HTML.gif , we denote
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_Equ24_HTML.gif
(220)
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_Equ25_HTML.gif
(221)
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_Equ26_HTML.gif
(222)
and let
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_Equ27_HTML.gif
(223)
where
http://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
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_Equ29_HTML.gif
(225)
Moreover, if we let http://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
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq51_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq52_HTML.gif ,
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_Equ31_HTML.gif
(227)
Therefore, there exists a http://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq53_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq54_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq55_HTML.gif converge uniformly to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq56_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq57_HTML.gif and
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_Equ32_HTML.gif
(228)
Since http://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
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_Equ33_HTML.gif
(229)
So,
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_Equ34_HTML.gif
(230)
which shows that http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq60_HTML.gif , there exist http://static-content.springer.com/image/art%3A10.1155%2F2009%2F685040/MediaObjects/13661_2009_Article_871_IEq61_HTML.gif such that
http://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

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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.