- Research Article
- Open Access
© F. Xu and Z. Meng. 2009
Received: 25 February 2009
Accepted: 2 June 2009
Published: 5 July 2009
We study the following third-order -Laplacian -point boundary value problems on time scales , , , , , where is -Laplacian operator, that is, , , , . We obtain the existence of positive solutions by using fixed-point theorem in cones. In particular, the nonlinear term is allowed to change sign. The conclusions in this paper essentially extend and improve the known results.
The theory of time scales was initiated by Hilger  as a mean of unifying and extending theories from differential and difference equations. The study of time scales has lead to several important applications in the study of insect population models, neural networks, heat transfer, and epidemic models, see, for example [2–6]. Recently, the boundary value problems with -Laplacian operator have also been discussed extensively in literature; for example, see [7–18]. However, to the best of our knowledge, there are not many results concerning the higher-order -Laplacian mutilpoint boundary value problem on time scales.
A time scale is a nonempty closed subset of . We make the blanket assumption that are points in . By an interval , we always mean the intersection of the real interval with the given time scale; that is .
author studied the existence of solutions for the nonlinear boundary value problem by using Krasnoselskii's fixed point theorem and Leggett and Williams fixed point theorem, respectively.
They established a corresponding iterative scheme for the problem by using the monotone iterative technique.
All the above works were done under the assumption that the nonlinear term is nonnegative. The key conditions used in the above papers ensure that positive solution is concave down. If the nonlinearity is negative somewhere, then the solution needs no longer to be concave down. As a result, it is difficult to find positive solutions of the -Laplacian equation when the nonlinearity changes sign. In particular, little work has been done on the existence of positive solutions for higher order -Laplacian -point boundary value problems with nonlinearity being nonnegative on time scales. Therefore, it is a natural problem to consider the existence of positive solution for higher order -Laplacian equations with sign changing nonlinearity on time scales. This paper attempts to fill this gap in literature.
2. Preliminaries and Lemmas
for all . If , is said to be right scattered, 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 .
By caculating, we can easily get (2.7). So we omit it.
The proof is completed.
This completes the proof.
Lemma 2.11 (see ).
3. Main Results
We now give our results on the existence of positive solutions of BVP (1.6).
Then, the BVP (1.6) has at least one positive solution.
Then, the BVP (1.6) has at least two positive solutions.
4. An Example
In the section, we present some simple examples to explain our results.
This project was supported by the National Natural Science Foundation of China (10471075, 10771117).
- 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
- Agarwal RP, O'Regan D: Nonlinear boundary value problems on time scales. Nonlinear Analysis: Theory, Methods & Applications 2001,44(4):527–535. 10.1016/S0362-546X(99)00290-4MATHMathSciNetView ArticleGoogle Scholar
- Atici FM, Guseinov GSh: On Green's functions and positive solutions for boundary value problems on time scales. Journal of Computational and Applied Mathematics 2002,141(1–2):75–99. 10.1016/S0377-0427(01)00437-XMATHMathSciNetView ArticleGoogle 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
- Bohner M, Peterso A (Eds): Advances in Dynamic Equations on Time Scales. Birkhäuser, Boston, Mass, USA; 2003:xii+348.MATHGoogle Scholar
- Sun HR, Li WT: Positive solutions for nonlinear -point boundary value problems on time scales. Acta Mathematica Sinica 2006,49(2):369–380.MATHMathSciNetGoogle Scholar
- Sun H-R, Li W-T: Existence theory for positive solutions to one-dimensional -Laplacian boundary value problems on time scales. Journal of Differential Equations 2007,240(2):217–248. 10.1016/j.jde.2007.06.004MATHMathSciNetView ArticleGoogle Scholar
- Su Y-H, Li W-T, Sun H-R: Triple positive pseudo-symmetric solutions of three-point BVPs for -Laplacian dynamic equations on time scales. Nonlinear Analysis: Theory, Methods & Applications 2008,68(6):1442–1452.MATHMathSciNetView ArticleGoogle 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
- He Z, Jiang X: Triple positive solutions of boundary value problems for -Laplacian dynamic equations on time scales. Journal of Mathematical Analysis and Applications 2006,321(2):911–920. 10.1016/j.jmaa.2005.08.090MATHMathSciNetView ArticleGoogle Scholar
- Xu FY: Positive solutions for third-order nonlinear -Laplacian -point boundary value problems on time scales. Discrete Dynamics in Nature and Society 2008, 2008:-16.Google Scholar
- Su YH, Li S, Huang C: Positive solution to a singular -Laplacian BVPs with sign-changing nonlinearity involving derivative on time scales. Advances in Difference Equations 2009, 2009:-21.Google Scholar
- Su YH, Li WT: Existence of positive solutions to a singular -Laplacian dynamic equations with sign changing nonlinearity. Acta Mathematica Scientia 2009, 52: 181–196.MATHMathSciNetGoogle Scholar
- Xu FY: Positive solutions for multipoint boundary value problems with one-dimensional -Laplacian operator. Applied Mathematics and Computation 2007,194(2):366–380. 10.1016/j.amc.2007.04.118MATHMathSciNetView ArticleGoogle Scholar
- Su YH, Li WT: Existence of positive solutions to a singular -Laplacian dynamic equations with sign changing nonlinearity. Acta Mathematica Scientia 2008, 28: 51–60.MathSciNetGoogle Scholar
- Su Y-H: Multiple positive pseudo-symmetric solutions of -Laplacian dynamic equations on time scales. Mathematical and Computer Modelling 2009,49(7–8):1664–1681. 10.1016/j.mcm.2008.10.010MATHMathSciNetView ArticleGoogle Scholar
- Su Y-H, Li W-T, Sun H-R: Positive solutions of singular -Laplacian BVPs with sign changing nonlinearity on time scales. Mathematical and Computer Modelling 2008,48(5–6):845–858. 10.1016/j.mcm.2007.11.008MATHMathSciNetView ArticleGoogle Scholar
- Zhou C, Ma D: Existence and iteration of positive solutions for a generalized right-focal boundary value problem with -Laplacian operator. Journal of Mathematical Analysis and Applications 2006,324(1):409–424. 10.1016/j.jmaa.2005.10.086MATHMathSciNetView ArticleGoogle Scholar
- Anderson DR: Green's function for a third-order generalized right focal problem. Journal of Mathematical Analysis and Applications 2003,288(1):1–14. 10.1016/S0022-247X(03)00132-XMATHMathSciNetView ArticleGoogle Scholar
- Lan KQ: Multiple positive solutions of semilinear differential equations with singularities. Journal of the London Mathematical Society 2001,63(3):690–704. 10.1112/S002461070100206XMATHMathSciNetView ArticleGoogle Scholar
- Guo DJ, 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
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.