Existence of positive solution for a third-order three-point BVP with sign-changing Green ’ s function ∗

By using the Guo-Krasnoselskii fixed point theorem, we investigate the following thirdorder three-point boundary value problem


Introduction
Third-order differential equations arise from a variety of different areas of applied mathematics and physics, e.g., in the deflection of a curved beam having a constant or varying cross section, a three-layer beam, electromagnetic waves or gravity driven flows and so on [3].
Recently, the existence of single or multiple positive solutions to some third-order three-point boundary value problems (BVPs for short) has received much attention from many authors, see [1,2,5,12,15,16] and the references therein.
However, all the above-mentioned papers are achieved when the corresponding Green's functions are positive, which is a very important condition.A natural question is that whether we can obtain the existence of positive solutions to some third-order three-point BVPs when the corresponding Green's functions are sign-changing.
In 2008, Palamides and Smyrlis [11] studied the existence of at least one positive solution to the singular third-order three-point BVP with an indefinitely signed Green's function where η ∈ 17 24 , 1 .Their technique was a combination of the Guo-Krasnoselskii fixed point theorem and properties of the corresponding vector field.
In 2012, by using the Guo-Krasnoselskii and Leggett-Williams fixed point theorems, Sun and Zhao [13,14] discussed the third-order three-point BVP with sign-changing Green's function where η ∈ ( 1 2 , 1).They obtained the existence of single or multiple positive solutions to the BVP (1.1) and proved that the obtained solutions were concave on [0, η] and convex on [η, 1].
It is worth mentioning that there are other type of works on sign-changing Green's functions which prove the existence of sign-changing solutions, positive in some cases, see Infante and Webb's papers [6][7][8].EJQTDE, 2013 No. 30, p. 2 In this paper we study the following third-order three-point BVP Throughout this paper, we always assume that α ∈ [0, 2) and η ∈ [ , 1).Obviously, the BVP (1.1) is a special case of the BVP (1.2).However, it is necessary to point out that this paper is not a simple extension of [13].In fact, if we let α = 0, then η ∈ [ 1 2 , 1), which is different from the restriction in [13].On the other hand, compared with [13], we can only prove that the obtained solution is concave on [0, η].
Our main tool is the following well-known Guo-Krasnoselskii fixed point theorem [4,9]: Then T has a fixed point in K ∩ (Ω 2 \ Ω 1 ).

Preliminaries
For the BVP we have the following lemma.

Lemma 2.1 The BVP (2.1) has only trivial solution.
Proof.It is simple to check.

Main results
For convenience, we denote Then it is obvious that 0 < B < A.
) is continuous and satisfies the following conditions: (H1) For each u ∈ [0, +∞), the mapping t → f (t, u) is decreasing; (H2) For each t ∈ [0, 1], the mapping u → f (t, u) is increasing; (H3) There exist two positive constants r and R with r = R such that Then the BVP (1.2) has a positive and decreasing solution u satisfying min{r, R} ≤ u ≤ max{r, R}.Moreover, the obtained solution u(t) is concave on [0, η].
Then it is easy to see that K is a cone in C [0, 1].Now, we define an operator T on K by Obviously, if u is a fixed point of T in K, then u is a nonnegative and decreasing solution of the BVP (1.2).In what follows, we will seek a fixed point of T in K by using Theorem 1.1.
First, by Lemma 2.2 and Lemma 2.3, we know that T : K → K. Furthermore, although G(t, s) is not continuous, it follows from known textbook results, for example see [10], that Next, for any u ∈ K, we claim that Now, without loss of generality, we assume that r < R. Let For any u ∈ K ∩ ∂Ω 1 , we get 0 ≤ u(s) ≤ r for s ∈ [0, 1], which together with (H3) implies This shows that