Positive Solutions of Second-order Three-point Boundary Value Problems with Sign-changing Coefficients

In this article, we investigate the boundary-value problem x (t) + h(t) f (x(t)) = 0, t ∈ [0, 1], x(0) = βx (0), x(1) = x(η), changes sign on [0, 1]. By the Guo–Krasnosel'ski˘ ı fixed-point theorem in a cone, the existence of positive solutions is obtained via a special cone in terms of superlinear or sublinear behavior of f .


Introduction
For the first time Liu [7] considered the existence of positive solutions to the following secondorder three-point boundary value problems x(0) = 0, x(1) = δx(η), (1.1) where λ is a positive parameter, η ∈ (0, 1), f ∈ C([0, ∞), [0, ∞)) is nondecreasing, δ ∈ (0, 1) and h(t) is continuous and especially changes sign on [0, 1] which is different from the nonnegative assumption in most of these studies.Karaca [4] studied the problems with more general boundary conditions Corresponding author.Email: gwzhang@mail.neu.edu.cn,gwzhangneum@sina.comwhere α ≥ 0, β ≥ 0, α + β > 0 with 0 < δ < 1, f , h as in (1.1).The authors of [4,7] showed the existence of at least one positive solution by applying the fixed-point theorem in a cone.Similar methods for a different problem are in [9].Let E be a Banach space, the nonempty subset P is called a cone in E if it is a closed convex set and satisfies the properties that λx ∈ P for any λ > 0, x ∈ P and that ±x ∈ P implies x = 0 (the zero element in E) (see [3]).
A question is whether one can have boundary condition x(1) = δx(η) with δ < (β + 1)/(β + η) in problem (1.2) with α = 1, which is the necessary condition when f ≥ 0. We only consider one (less complicated) special case δ = 1.If α = 0, the corresponding linear problem for g ∈ C[0, 1] will be which is a resonance problem.So it is acceptable that α > 0 and may be supposed to be α = 1.For that reason, we investigate the existence of positive solutions to the three-point boundary-value problem where is continuous and is sign changing on [0, 1].The existence of positive solutions is obtained via a special cone (see (2.5)) in terms of superlinear or sublinear behavior of f by the Guo-Krasnosel'skiȋ fixed-point theorem in a cone.The ideas here are similar to the papers [4,7] and [9], but note that the signs on h are opposite to those in [4,7].Other relevant research can be seen in [1,2,5,8,10].

Preliminaries
We will use the following assumptions.
) is continuous and nondecreasing.
where τ and A are as in (H 3 ).
Proof.By Lemma 2.4 we have x = max t∈[η,1] x(t) and denote Therefore, and hence where τ is as in (H 3 ).
By the same way, the other inequality holds.

Main results
For x ∈ P define the operator T as the following: where G(t, s) is in (2.4).
) is convex on [0, η] and is concave on [η, 1] respectively.These mean that T : P → P. At last, we know that T is completely continuous from the Arzelà-Ascoli theorem.
It follows from Lemma 2.2 that there exists a positive solution to (1.4) if and only if T has a fixed point in P. In order to prove the existence of positive solution we need the following Guo-Krasnosel'skiȋ fixed point theorem in the cone [3,6].Lemma 3.2.Let E be a Banach space and P be a cone in E. Suppose that Ω 1 and Ω 2 are bounded open sets in E with 0 ∈ Ω 1 and Ω 1 ⊂ Ω 2 .If T : P ∩ (Ω 2 \Ω 1 ) → P is a completely continuous operator and satisfies either (i) Tx ≤ x for x ∈ P ∩ ∂Ω 1 and Tx ≥ x for x ∈ P ∩ ∂Ω 2 ; or (ii) Tx ≥ x for x ∈ P ∩ ∂Ω 1 and Tx ≤ x for x ∈ P ∩ ∂Ω 2 , then T has a fixed point in P ∩ (Ω 2 \Ω 1 ).
then (1.4) has at least one positive solution.