Existence of three symmetric positive solutions for a second-order multi-point boundary value problem on time scales
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RESEARCH
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Existence of three symmetric positive solutions for a second-order multi-point boundary value problem on time scales Aycan Sinanoglu1 , Ilkay Y Karaca1* , Fatma Tokmak1,2 and Tugba Senlik1 * Correspondence: [email protected] 1 Department of Mathematics, Ege University, Bornova, Izmir, 35100, Turkey Full list of author information is available at the end of the article
Abstract 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 Keywords: boundary value problem; fixed point theorem; symmetric positive solutions; time scales
1 Introduction 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 [, ], and Lakshmikantham et al. [] and the references therein. The study of multi-point linear boundary value problems was initiated by II’in and Moiseev [, ]. 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 [–] 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 [, ]. 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 [–]. Yao [] studied the following boundary value problem (BVP):
x (t) + ω(t)f (x(t)) = , t ∈ (, ), αx() – βx () = , αx() + βx () = .
©2014 Sinanoglu et al.; licensee Springer. 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 cited.
Sinanoglu et al. Advances in Difference Equations 2014, 2014:81 http://www.advancesindifferenceequations.com/content/2014/1/81
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This author obtained the existence of n symmetric positive solutions and established a corresponding iterative scheme by using a monotone iterative techniq
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