Triple positive solutions for second-order four-point boundary value problem

In this paper, we study the existence of triple positive solutions for second-order four-point boundary value problem with sign changing nonlinearities. We first study the associated Green’s function and obtain some useful properties. Our main tool is the fixed point theorem due to Avery and Peterson. The results of this paper are new and extent previously known results.


Introduction
Boundary value problems (BVP) with sign changing nonlinearities have received special attention from many authors in recent years.Recently, existence results for positive solutions of second-order three-point boundary value problems with sign changing nonlinearities have been studied by some authors (see [3][4][5][6]).In [4], by applying the Krasnoseiskii fixed-point theorem in a cone, Liu have proved the existence of at least one positive solution for the following second-order three-point BVP u (t) + λa(t)f (u(t)) = 0, t ∈ (0, 1), u(0) = 0, u(1) = βu(η).
Inspired by [2], in this paper, by using the fixed point theorem due to Avery and Peterson, we consider the following second-order four-point boundary value problem with sign changing nonlinearities We make the following assumptions: . Moreover, a(t) does not vanish identically on any subinterval of [0, 1]; (H 3 ) There exists constant τ 1 ∈ (βη, η), such that where a + (t) = max{a(t), 0}, a − (t) = − min{a(t), 0} and The aim of this paper is to improve the previous existence results in [4].By using the fixed point theorem due to Avery and Peterson, we will study the existence of triple positive solutions for the BVP (1.3) under some conditions concerning the function a that is sign-changing on [0, 1].To the best of our knowledge, to date no paper has appeared in the literature which discusses the existence of positive solutions for the BVP (1.3).This paper attempts to fill this gap in the literature.

Preliminary lemmas
Let ϑ and θ be nonnegative continuous convex functionals on K, κ be a nonnegative continuous concave functional on K, and ψ be a nonnegative continuous functional on K. Then for positive real numbers a, b, c and d, we define the following convex sets: Lemma 2.1 [1].Let K be a cone in a Banach space E. Let ϑ and θ be nonnegative continuous convex functionals on K, κ be a nonnegative continuous concave functional on K, and ψ be a nonnegative continuous functional on K satisfying ψ(λu) ≤ λψ(u) for 0 ≤ λ ≤ 1, such that for some positive numbers M and d, for all u ∈ K(ϑ, d).Suppose T : K(ϑ, d) → K(ϑ, d) is completely continuous and there exists positive numbers a, b and c with a < b such that (C 3 ) 0 / ∈ K(ϑ, ψ, a, d) and ψ(T u) < a for u ∈ K(ϑ, ψ, a, d), with ψ(u) = a.
Then T has at least three fixed points u 1 , u 2 and has a unique solution Proof.The proof follows by direct calculations.
Let G(t, s) be the Green's function for the BVP (2.2)-(2.3).By direct calculation, we have where Lemma 2.3.Suppose (H 0 ) holds.Then G(t, s) has the following properties where τ 1 and Λ are given as in (H 3 ).
Proof.(i) By calculating, we obtain From this and (H 0 ), we have Thus, (i) holds.
(ii) By (2.4) and (H 0 ), we have Thus, If s 1 ≤ t ≤ s 2 , we have by (2.4) and (H 0 ) that So, Then, This completes the proof.
Let C[0, 1] be the Banach space with norm u = max and define the cone K ⊂ X and the operator T : where . Thus, we have and So, (2.5) holds.
Proof.Firstly, we claim T (K) ⊂ K.In fact, for all u ∈ K, we have by Lemma 2.6 that This show that T : K → K.It can be shown that T : K → K is completely continuous by Arzela-Ascoli theorem.

Main result
Let the nonnegative continuous concave functional κ on K, the nonnegative continuous convex functionals θ, ϑ and the nonnegative continuous functional ψ be defined on the cone K by For the sake of brevity, we denote where G(τ 2 , s)a + (s)ds.