Existence and Nonexistence of Positive Solutions for a Higher-Order Three-Point Boundary Value Problem

and Applied Analysis 3 (c) β(u + V) ⩾ β(u) + β(V) for all u, V ∈ P, β(0) = 0, β(u) ̸ = 0 if u ̸ = 0. The approach used in proving the existence results in this paper is the following fixed point theorem of cone expansion and compression of functional type due to Avery et al. [21]. Theorem 4. Let 1 and 2 be two bounded open sets in a Banach space E such that 0 ∈ 1 andΩ1 ⊆ 2 and P is a cone inE. SupposeT : 21 → P is a completely continuous operator, α and γ are nonnegative continuous functional on P, and one of the two conditions: (K1) α satisfies Property A1 with α(Tu) ⩾ α(u), for all u ∈ P∩∂Ω1, and γ satisfies PropertyA2with γ(Tu) ⩽ γ(u), for all u ∈ P ∩ ∂Ω2, or (K2) γ satisfies Property A2 with γ(Tu) ⩽ γ(u), for all u ∈ P∩∂Ω1, and α satisfies PropertyA1 with α(Tu) ⩾ α(u), for all u ∈ P ∩ ∂Ω2, is satisfied, then T has at least one fixed point in P∩ 2 \Ω1). 3. Expression and Properties of Green’s Function In this section we present the expression and properties of Green’s function associated with BVP (1). We will consider the Banach space E = C[0, 1] equipped with norm ‖u‖ = max0⩽t⩽1|u(t)|. In arriving our result, we need the following two preliminary lemmas. Lemma 5. Let h ∈ C[0, 1], then boundary value problem u (n) (t) + h (t) = 0, 0 ⩽ t ⩽ 1, u (i) (0) = 0, i = 0, 1, . . . , n − 2, u (p) (1) = ξu (p) (η) (8) has a unique solution

In recent years, the existence and multiplicity of positive solutions for nonlinear higher-order ordinary differential equations with three-point boundary conditions have been studied by several authors; we can refer to [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19] and the references therein.For example, Eloe and Ahmad in [4] discussed the existence of positive solutions of a nonlinear th-order three-point boundary value problem where 0 <  < 1, 0 <  −1 < 1.The existence of at least one positive solution if  is either superlinear or sublinear was established by applying the fixed point theorem in cones due to Krasnosel'skii.Under conditions different from those imposed in [4], Graef and Moussaoui in [5] studied the existence of both sign changing solutions and positive solutions for BVP (2).Hao et al. [6] are devoted to the existence and multiplicity of positive solutions for BVP (2) under certain suitable weak conditions, where () may be singular at  = 0 and/or  = 1.The main tool used is also the Krasnosel'skii fixed point theorem.Graef et al. (3) where  ⩾ 4 is an integer and  ∈ (1/2, 1) is a constant.Sufficient conditions for the existence, nonexistence, and multiplicity of positive solutions of this problem are obtained by using Krasnosel'skii's fixed point theorem, Leggett-Williams' fixed point theorem, and the five-functional fixed point theorem.
It is well known that fixed point theorems have been applied to various boundary value problems to show the existence and multiplicity of positive solutions.Fixed point theorems and their applications to nonlinear problems have a long history; the recent book by Agarwal et al. [20] contains an excellent summary of the current results and applications.Recently, Avery et al. [21] generalized the fixed point theorem of cone expansion and compression of norm type by replacing the norms with two functionals satisfying certain conditions to produce a fixed point theorem of cone expansion and compression of functional type, and then they applied the fixed point theorem to verify the existence of a positive solution to a second order conjugate boundary value problem.
Motivated greatly by the above-mentioned works, in this paper we will try using this new fixed point theorem to consider the existence of monotone positive solution to BVP (1).The methods used to prove the existence results are standard; however, their exposition in the framework of BVP (1) is new.This paper is organized as follows.In Section 2, we present some definitions and background results on cones and completely continuous operators.We also state the fixed point theorem of cone expansion and compression of functional type due to Avery, Henderson, and O'Regan.Expression and properties of Green's function will be given in Section 3. The main results will be given in Section 4.

Preliminaries
In this section, for the convenience of the reader, we present some definitions and background results on cones and completely continuous operators.We also state a fixed point theorem of cone expansion and compression of functional type due to Avery, Henderson, and O'Regan.
Every cone  ⊂  induces an ordering in  given by  ⩽ V if and only if V −  ∈ .Definition 2. Let  be a real Banach space.An operator  :  →  is said to be completely continuous if it is continuous and maps bounded sets into precompact sets.Definition 3. A map  is said to be a nonnegative continuous concave functional on a cone  of a real Banach space  if  :  → [0, +∞) is continuous and Similarly we said the map  is a nonnegative continuous convex functional on a cone  of a real Banach space  if  :  → [0, +∞) is continuous and We say the map  is sublinear functional if All the concepts discussed above can be found in [22].
Property A2.Let  be a cone in a real Banach space  and Ω be a bounded open subset of  with 0 ∈ Ω.Then a continuous functional  :  → [0, ∞) is said to satisfy Property A2 if one of the following conditions holds: The approach used in proving the existence results in this paper is the following fixed point theorem of cone expansion and compression of functional type due to Avery et al. [

Expression and Properties of Green's Function
In this section we present the expression and properties of Green's function associated with BVP (1).We will consider the Banach space  = [0, 1] equipped with norm ‖‖ = max 0⩽⩽1 |()|.In arriving our result, we need the following two preliminary lemmas.