Existence of three symmetric positive solutions for a second-order multi-point boundary value problem on time scales
© Sinanoglu et al.; licensee Springer. 2014
Received: 11 October 2013
Accepted: 21 February 2014
Published: 12 March 2014
In this article, we investigate the existence of at least three symmetric positive solutions of a second-order multi-point boundary value problem on time scales. The ideas involve Bai and Ge’s fixed point theorem. As an application, we give an example to demonstrate our main result.
MSC:34B10, 34B18, 39A10.
Calculus on time scales was introduced by Hilger  as a theory which includes both differential and difference calculus as a special cases. In the past few years, it has found a considerable amount of interest and attracted the attention of many researchers. Time scale calculus would allow the exploration of a variety of situations in economic, biological, heat transfer, stock market, and epidemic models; see the monographs of Aulbach and Hilger , Bohner and Peterson [3, 4], and Lakshmikantham et al.  and the references therein.
The study of multi-point linear boundary value problems was initiated by II’in and Moiseev [6, 7]. Since then, the more general nonlinear multi-point boundary value problems have been widely studied by many authors. The multi-point boundary value problems arise frequently in applied mathematics and physics, see for instance [8–16] and the references therein. At the same time, interest in obtaining the solutions on time scales has been on-going for several years.
On the other hand, the existence of symmetric positive solutions of second-order boundary value problems have been studied by some authors, see [17, 18]. Most of the study of the symmetric positive solution is limited to the Dirichlet boundary value problem, the Sturm-Liouville boundary value problem and the Neumann boundary value problem. However, there is not so much work on symmetric positive solutions for second-order m-point boundary value problems; see [19–21].
This author obtained the existence of n symmetric positive solutions and established a corresponding iterative scheme by using a monotone iterative technique.
By using the Leggett-Wiliams fixed point theorem and the coincidence degree theorem of Mawhin, he studied the existence of three positive solutions for a multi-point boundary value problem.
where be a symmetric bounded time scale, with , , such that and is called the quasi-Δ-derivative of .
(C1) , , , , ;
(C2) , , with , such that , for .
By using Bai and Ge’s fixed point theorem , we get the existence of at least three symmetric positive solutions for the BVP (1.1). In fact, our results are new when (the differential case) and . Hence, our new results naturally complement recent advances in the literature.
This paper is organized as follows. In Section 2, we provide some preliminary lemmas which are key tools for our main results. We give and prove our main results in Section 3. Finally, in Section 4, we give an example to demonstrate our results.
In this section, we present auxiliary lemmas which will be used later.
where , are given in (2.3) and (2.4), respectively, and .
Similarly, we can see that the other boundary condition is satisfied. □
Lemma 2.2 For , we have .
Similarly, we can prove that , .
So, we have for all , i.e., is a symmetric function on . □
The proof is complete. □
Lemma 2.4 Suppose that (C1), (C2) hold, then where .
Proof We have from (2.5) for .
The proof is complete. □
Lemma 2.5 For , .
Proof One can easily see that the inequality holds. □
Lemma 2.6 Let (C1), (C2) hold. Then is completely continuous.
i.e., for . Therefore, is symmetric on .
So, and then . Next, by standard methods and the Arzela-Ascoli theorem, one can easily prove that the operator T is completely continuous. □
3 Main results
We are now ready to apply the fixed point theorem due to Bai and Ge  to the operator T in order to get sufficient conditions for the existence of multiple positive solutions to the problem (1.1).
With (3.1) and (3.2), Ω is a bounded nonempty open subset in P.
To prove our results, we need the following fixed point theorem due to Bai and Ge in .
Theorem 3.1 
(A1) , for ,
(A2) , , for all ,
(A3) , for all with .
Then on the cone P, ρ is a concave functional, and μ and γ are convex functionals satisfying (3.1) and (3.2).
(S1) , for ,
(S2) , for ,
(S3) , for .
Proof The problem (1.1) has a solution if and only if y satisfies the operator equation . Thus we set out to verify that the operator T satisfies all conditions of Theorem 3.1. The proof is divided into four steps.
Step 2. We show that condition (A1) in Theorem 3.1 holds.
Consequently, condition (A1) of Theorem 3.1 is satisfied.
Step 3. We now show that condition (A2) in Theorem 3.1 is satisfied. In the same way as in Step 1, if , then assumption (S1) yields for . Therefore, condition (A2) of Theorem 3.1 is satisfied.
The proof is complete. □
To illustrate how our main result can be used in practice we present an example.
The authors would like to thank the referees for their valuable suggestions and comments.
- Hilger S: Analysis on measure chains-a unified approach to continuous and discrete calculus. Results Math. 1990, 18: 18–56. 10.1007/BF03323153MathSciNetView ArticleGoogle Scholar
- Aulbach B, Hilger S: Linear dynamic processes with inhomogeneous time scale. 59. In Nonlinear Dynamics and Quantum Dynamical Systems(Gaussig, 1990). Akademie Verlag, Berlin; 1990:9–20.Google Scholar
- Bohner M, Peterson A: Dynamic Equations on Time Scales: An Introduction with Applications. Birkhäuser, Boston; 2001.View ArticleGoogle Scholar
- Bohner M, Peterson A: Advances in Dynamic Equations on Time Scales. Birkhäuser, Boston; 2003.View ArticleGoogle Scholar
- Lakshmikantham V, Sivasundaram S, Kaymakcalan B: Dynamic Systems on Measure Chains. Kluwer Academic, Dordrecht; 1996.View ArticleGoogle Scholar
- Il’in VA, Moiseev EI: A nonlocal boundary value problem of the first kind for the Sturm-Liouville operator in differential and difference interpretations. Differ. Uravn. 1987, 23: 1198–1207.MathSciNetGoogle Scholar
- Il’in VA, Moiseev EI: A nonlocal boundary value problem of the second kind for the Sturm-Liouville operator. Differ. Uravn. 1987, 23: 1422–1431.MathSciNetGoogle Scholar
- Feng W: On an m -point boundary value problem. Nonlinear Anal. 1997, 30: 5369–5374. 10.1016/S0362-546X(97)00360-XMathSciNetView ArticleGoogle Scholar
- Gupta CP: Solvability of a three-point nonlinear boundary value problem for a second order ordinary differential equation. J. Math. Anal. Appl. 1992, 168: 540–551. 10.1016/0022-247X(92)90179-HMathSciNetView ArticleGoogle Scholar
- Kosmatov N: Multi-point boundary value problems on an unbounded domain at resonance. Nonlinear Anal. 2008, 68: 2158–2171. 10.1016/j.na.2007.01.038MathSciNetView ArticleGoogle Scholar
- Ma R: Multiple positive solutions for nonlinear m -point boundary value problems. Appl. Math. Comput. 2004, 148: 249–262. 10.1016/S0096-3003(02)00843-3MathSciNetView ArticleGoogle Scholar
- Sedziwy S: Multipoint boundary value problems for a second-order ordinary differential equation. J. Math. Anal. Appl. 1999, 236: 384–398. 10.1006/jmaa.1999.6441MathSciNetView ArticleGoogle Scholar
- Sun HR: Triple positive solutions for p -Laplacian m -point boundary value problem on time scales. Comput. Math. Appl. 2009, 58: 1736–1741. 10.1016/j.camwa.2009.07.083MathSciNetView ArticleGoogle Scholar
- Su YH: Arbitrary positive solutions to a multi-point p -Laplacian boundary value problem involving the derivative on time scales. Math. Comput. Model. 2011, 53: 1742–1747. 10.1016/j.mcm.2010.12.052View ArticleGoogle Scholar
- Dogan A: Existence of three positive solutions for an m -point boundary-value problem on time scales. Electron. J. Differ. Equ. 2013., 2013: Article ID 149Google Scholar
- Dogan A: Existence of multiple positive solutions for p -Laplacian multipoint boundary value problems on time scales. Adv. Differ. Equ. 2013., 2013: Article ID 238Google Scholar
- Avery RI, Henderson J: Three symmetric positive solutions for a second order boundary value problem. Appl. Math. Lett. 2000, 13: 1–7.MathSciNetView ArticleGoogle Scholar
- Henderson J, Thompson HB: Multiple symmetric positive solutions for a second order boundary value problem. Proc. Am. Math. Soc. 2000, 128: 2373–2379. 10.1090/S0002-9939-00-05644-6MathSciNetView ArticleGoogle Scholar
- Fang H: Existence of symmetric positive solutions for m -point boundary value problems for second-order dynamic equations on time scales. Math. Theory Appl. 2008, 228: 65–68.Google Scholar
- Lui X: Existence and multiplicity of symmetric positive solutions for singular second-order m -point boundary value problem. Int. J. Math. Anal. 2011, 5: 1453–1457.Google Scholar
- Sun Y, Zhang X: Existence of symmetric positive solutions for an m -point boundary value problem. Bound. Value Probl. 2007., 2007: Article ID 79090Google Scholar
- Yao Q: Existence and iteration of n symmetric positive solutions for a singular two-point boundary value problem. Comput. Math. Appl. 2004, 47: 1195–1200. 10.1016/S0898-1221(04)90113-7MathSciNetView ArticleGoogle Scholar
- Kosmatov N: Symmetric solutions of a multi-point boundary value problem. J. Math. Anal. Appl. 2005, 309: 25–36. 10.1016/j.jmaa.2004.11.008MathSciNetView ArticleGoogle Scholar
- Kosmatov N: A symmetric solution of a multipoint boundary value problem at resonance. Abstr. Appl. Anal. 2006., 2006: Article ID 54121Google Scholar
- Bai Z, Ge W: Existence of three positive solutions for some second-order boundary value problems. Comput. Math. Appl. 2004, 48: 699–707. 10.1016/j.camwa.2004.03.002MathSciNetView ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.