- Open Access
Existence of concave symmetric positive solutions for a three-point boundary value problems
© Akcan and Hamal; licensee Springer. 2014
- Received: 10 September 2014
- Accepted: 17 November 2014
- Published: 9 December 2014
In this paper, we investigate the existence of triple concave symmetric positive solutions for the nonlinear boundary value problems with integral boundary conditions. The proof is based upon the Avery and Peterson fixed point theorem. An example which supports our theoretical result is also indicated.
MSC:34B10, 39B18, 39A10.
- three-point boundary value problem
- symmetric positive solution
- integral boundary condition
- fixed point theorem
By a symmetric positive solution of BVP (1.1) we mean a symmetric function such that for and satisfies BVP (1.1).
Recently, many authors have focused on the existence of symmetric positive solutions for ordinary differential equation boundary value problems; for example, see [1–5] and the references therein. However, multi-point boundary value problems included the most recent works [1–4, 6–9] and boundary value problems with integral boundary conditions for ordinary differential equations have been studied by many authors; one may refer to [5, 10–12]. Motivated by the works mentioned above, we aim to investigate existence results for concave symmetric positive solutions of BVP (1.1) by applying the fixed point theorem of Avery and Peterson.
The organization of this paper is as follows. Section 2 of this paper contains some preliminary lemmas. In Section 3, by applying the Avery and Peterson fixed point theorem, we obtain concave symmetric positive solutions for BVP (1.1). In Section 4, an example will be presented to illustrate the applicability of our result.
Throughout this paper, we always assume that the following assumption is satisfied:
(H1) , for , and for all .
In this section, we present several lemmas that will be used in the proof of our result.
The proof is complete. □
The functions H and G have the following properties.
Proof From the definition of , , and , we have . □
Lemma 2.3 , for .
Proof From the definition of , we get and for . □
Lemma 2.4 If and we let , then the unique solution u of BVP (1.1) satisfies for .
Proof From the definition of , Lemma 2.2, Lemma 2.3, and , we have . □
Let . Then E is Banach space with the norm , where .
where and are given by (2.4) and (2.5). Clearly, u is the solution of BVP (1.1) if and only if u is a fixed point of the operator T.
Now we divide the proof into two cases.
where , these equations complete the proof. □
Now we divide the proof into two cases.
where , and for , one can easily see that . These equations complete the proof. □
In this section, we will apply the Avery-Peterson fixed point theorem to establish the existence of at least three concave symmetric positive solutions of BVP (1.1).
Now we state the Avery-Peterson fixed point theorem.
Lemma 3.1 Assume that (H1) is satisfied and let . Then the operator T is completely continuous.
These equations imply that the operator T is uniformly bounded. Now we show that Tu is equi-continuous.
We separate these three conditions:
Case (i). ;
Case (ii). ;
Case (iii). .
So, we see that Tu is equi-continuous. By applying the Arzela-Ascoli theorem, we can guarantee that is relatively compact, which means T is compact. Then we find that T is completely continuous. This completes the proof. □
Proof BVP (1.1) has a solution if and only if u solves the operator equation . So we need to verify that operator T satisfies the Avery-Peterson fixed point theorem, which will prove the existence of at least three fixed points of T.
Then in the cone P, θ and γ are convex as α is concave. It is well known that for all and . Moreover, from Lemma 2.5, . Now, we will prove the main theorem in four steps.
Step 1. We will show that .
Then . Therefore .
Thus, and .
This shows that condition (S1) of Theorem 3.1 is satisfied.
so condition (S2) holds.
Therefore, our proof is complete. □
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