- Research Article
- Open Access
Existence Results for Higher-Order Boundary Value Problems on Time Scales
© J. Liu and Y. Sang. 2009
Received: 22 March 2009
Accepted: 16 June 2009
Published: 29 July 2009
Time scales and time-scale notationare introduced well in the fundamental texts by Bohner and Peterson [1, 2], respectively, as important corollaries. In, the recent years, many authors have paid much attention to the study of boundary value problems on time scales (see, e.g., [3–17]). In particular, we would like to mention some results of Anderson et al. [3, 5, 6, 14, 16], DaCunha et al. , and Agarwal and O'Regan , which motivate us to consider our problem.
with the same boundary conditions where is a positive parameter. They obtained some results for the existence of positive solutions by using the Krasnoselskii, the Schauder, and the Avery-Henderson fixed-point theorem.
and got many excellent results.
In this paper, by constructing one integral equation which is equivalent to the BVP (1.6) and (1.7), we study the existence of positive solutions. Our main tool of this paper is the following fixed-point index theorem.
Theorem 1.1 ().
The outline of the paper is as follows. In Section 2, for the convenience of the reader we give some definitions and theorems which can be found in the references, and we present some lemmas in order to prove our main results. Section 3 is developed in order to present and prove our main results. In Section 4 we present some examples to illustrate our results.
2. Preliminaries and Lemmas
Throughout we assume that are points in , and define the time-scale interval . In this paper, we also need the the following theorem which can be found in .
We will discuss it from three perspectives.
Sufficiency. Suppose that
For convenience, we set
3. The Existence of Positive Solution
Proof of Theorem 3.1.
Proof of Theorem 3.2.
Proof of Theorem 3.3.
In this section, in order to illustrate our results, we consider the following examples.
By Theorem 2.1, we have
By simple calculations, we have
The authors would like to thank the anonymous referee for his/her valuable suggestions, which have greatly improved this paper.
- Bohner M, Peterson A: Dynamic Equations on Time Scales, An Introduction with Applications. Birkhäuser, Boston, Mass, USA; 2001:x+358.MATHView ArticleGoogle Scholar
- Bohner M, Peterson A (Eds): Advances in Dynamic Equations on Time Scales. Birkhäuser, Boston, Mass, USA; 2003:xii+348.MATHGoogle Scholar
- Anderson DR, Karaca IY: Higher-order three-point boundary value problem on time scales. Computers & Mathematics with Applications 2008,56(9):2429–2443. 10.1016/j.camwa.2008.05.018MATHMathSciNetView ArticleGoogle Scholar
- DaCunha JJ, Davis JM, Singh PK: Existence results for singular three point boundary value problems on time scales. Journal of Mathematical Analysis and Applications 2004,295(2):378–391. 10.1016/j.jmaa.2004.02.049MATHMathSciNetView ArticleGoogle Scholar
- Anderson DR, Guseinov GSh, Hoffacker J: Higher-order self-adjoint boundary-value problems on time scales. Journal of Computational and Applied Mathematics 2006,194(2):309–342. 10.1016/j.cam.2005.07.020MATHMathSciNetView ArticleGoogle Scholar
- Anderson DR, Avery R, Davis J, Henderson J, Yin W: Positive solutions of boundary value problems. In Advances in Dynamic Equations on Time Scales. Birkhäuser, Boston, Mass, USA; 2003:189–249.View 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
- Kaufmann ER: Positive solutions of a three-point boundary-value problem on a time scale. Electronic Journal of Differential Equations 2003, (82):1–11.Google 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
- Boey KL, Wong PJY: Existence of triple positive solutions of two-point right focal boundary value problems on time scales. Computers & Mathematics with Applications 2005,50(10–12):1603–1620.MATHMathSciNetView ArticleGoogle Scholar
- Sun J-P: Existence of solution and positive solution of BVP for nonlinear third-order dynamic equation. Nonlinear Analysis: Theory, Methods & Applications 2006,64(3):629–636. 10.1016/j.na.2005.04.046MATHMathSciNetView ArticleGoogle Scholar
- Benchohra M, Henderson J, Ntouyas SK: Eigenvalue problems for systems of nonlinear boundary value problems on time scales. Advances in Difference Equations 2007, 2007:-10.Google Scholar
- Agarwal RP, Otero-Espinar V, Perera K, Vivero DR: Multiple positive solutions in the sense of distributions of singular BVPs on time scales and an application to Emden-Fowler equations. Advances in Difference Equations 2008, 2008:-13.Google Scholar
- Anderson DR: Oscillation and nonoscillation criteria for two-dimensional time-scale systems of first-order nonlinear dynamic equations. Electronic Journal of Differential Equations 2009,2009(24):13. Article ID 796851.Google Scholar
- Bohner M, Luo H: Singular second-order multipoint dynamic boundary value problems with mixed derivatives. Advances in Difference Equations 2006, 2006:-15.Google Scholar
- Anderson DR, Ma R: Second-order -point eigenvalue problems on time scales. Advances in Difference Equations 2006, 2006:-17.Google Scholar
- Henderson J, Peterson A, Tisdell CC: On the existence and uniqueness of solutions to boundary value problems on time scales. Advances in Difference Equations 2004,2004(2):93–109. 10.1155/S1687183904308071MATHMathSciNetView 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
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