- Research Article
- Open Access
© Changxiu Song. 2008
Received: 29 February 2008
Accepted: 25 June 2008
Published: 2 July 2008
This paper is concerned with the existence and nonexistence of positive solutions of the -Laplacian functional dynamic equation on a time scale, , , , , , . We show that there exists a such that the above boundary value problem has at least two, one, and no positive solutions for and , respectively.
Let be a closed nonempty subset of , and let have the subspace topology inherited from the Euclidean topology on . In some of the current literature, is called a time scale (please see [1, 2]). For notation, we will use the convention that, for each interval of will denote time-scale interval, that is,
-Laplacian problems with two-, three-, -point boundary conditions for ordinary differential equations and finite difference equations have been studied extensively, for example, see [1–4] and references therein. However, there are not many concerning the -Laplacian problems on time scales, especially for -Laplacian functional dynamic equations on time scales.
The motivations for the present work stems from many recent investigations in [5–10] and references therein. Especially, Kaufmann and Raffoul  considered a nonlinear functional dynamic equation on a time scale and obtained sufficient conditions for the existence of positive solutions, Li and Liu  studied the eigenvalue problem for second-order nonlinear dynamic equations on time scales. In this paper, our results show that the number of positive solutions of (1.1) is determined by the parameter . That is to say, we prove that there exists a such that (1.1) has at least two, one, and no positive solutions for and respectively.
If is said to be right scattered, and if is said to be left scattered. If is said to be right dense, and if is said to be left dense. If has a right-scattered minimum define otherwise set If has a left-scattered maximum define otherwise set
The following lemma is crucial to prove our main results.
Lemma 1.4 ().
2. Positive Solutions
It follows from (2.3) that the following lemma holds.
The proof of Lemma 2.1 can be found in .
By the Lebesgue dominated convergence theorem  together with (H3), it follows that decreases to a fixed point of the operator The proof is complete.
We need to prove that there exists a constant such that for all If the number of elements of is finite, then the result is obvious. If not, without loss of generality, we assume that there exists a sequence such that , where is the fixed point of the operator defined by (2.3) at
which is a contradiction. The proof is complete.
which is a contradiction. The proof is complete.
This work was supported by Grant 10571064 from NNSF of China, and by a grant from NSF of Guangdong.
- Avery R, Henderson J:Existence of three positive pseudo-symmetric solutions for a one-dimensional -Laplacian. Journal of Mathematical Analysis and Applications 2003, 277(2):395-404. 10.1016/S0022-247X(02)00308-6MATHMathSciNetView ArticleGoogle Scholar
- Liu Y, Ge W:Twin positive solutions of boundary value problems for finite difference equations with -Laplacian operator. Journal of Mathematical Analysis and Applications 2003, 278(2):551-561. 10.1016/S0022-247X(03)00018-0MATHMathSciNetView ArticleGoogle Scholar
- Cabada A:Extremal solutions for the difference -Laplacian problem with nonlinear functional boundary conditions. Computers & Mathematics with Applications 2001, 42(3–5):593-601.MATHMathSciNetView ArticleGoogle Scholar
- Wong F-H:Existence of positive solutions for -Laplacian boundary value problems. Applied Mathematics Letters 1999, 12(3):11-17. 10.1016/S0893-9659(98)00164-5MATHMathSciNetView ArticleGoogle Scholar
- Song C:Positive solutions of functional difference equations with -Laplacian operator. Advances in Difference Equations 2006, 2006:-9.Google 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
- Kaufmann ER, Raffoul YN: Positive solutions for a nonlinear functional dynamic equation on a time scale. Nonlinear Analysis: Theory, Methods & Applications 2005, 62(7):1267-1276. 10.1016/j.na.2005.04.031MATHMathSciNetView 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):1-14.MathSciNetGoogle Scholar
- He Z:Double positive solutions of three-point boundary value problems for -Laplacian dynamic equations on time scales. Journal of Computational and Applied Mathematics 2005, 182(2):304-315. 10.1016/j.cam.2004.12.012MATHMathSciNetView ArticleGoogle Scholar
- Li W-T, Liu X-L: Eigenvalue problems for second-order nonlinear dynamic equations on time scales. Journal of Mathematical Analysis and Applications 2006, 318(2):578-592. 10.1016/j.jmaa.2005.06.030MATHMathSciNetView 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
- 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
- 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
- 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
- Song C:Existence of solutions for -Laplacian functional dynamic equations on time scales. Electronic Journal of Differential Equations 2006, 2006(113):1-8.MATHGoogle Scholar
- Aulbach B, Neidhart L: Integration on measure chains. In Proceedings of the 6th International Conference on Difference Equations, July-August 2004, Augsburg, Germany. CRC; 239-252.Google Scholar
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