- Open Access
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.
- boundary value problem
- fixed point theorem
- symmetric positive solutions
- time scales
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. □
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.
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