Twin Positive Solutions for Three-point Boundary Value Problems of Higher-order Differential Equations

A new fixed point theorem on cones is applied to obtain the existence of at least two positive solutions of a higher-order three-point boundary value problem for the differential equation subject to a class of boundary value conditions. The associated Green's function is given. Some results obtained recently are generalized. 1. Introduction. The multipoint boundary value problems for ordinary differential equations arise in a variety of different areas of applied mathematics and physics. Linear and nonlinear second-order multipoint boundary value problems have been studied by several authors, we refer the reader and the references therein. Consider the nth-order two-point boundary value problem


Introduction.
The multipoint boundary value problems for ordinary differential equations arise in a variety of different areas of applied mathematics and physics.Linear and nonlinear second-order multipoint boundary value problems have been studied by several authors, we refer the reader to [6,7,8,9,10,11,12,13,14,15] and the references therein.
On the other hand, to the best of our knowledge, few authors have studied the existence of multiple positive solutions for higher-order multipoint boundary value problems.It is an interesting problem and one of the future research directions to discuss the solvability of the nth-order differential equations x (n) satisfying either k-point right focal boundary value conditions or k-point boundary value conditions [4,5].
Motivated by the results [1,2,3,4,5], we, in this paper, study the existence of multiple positive solutions for the nth-order three-point boundary value problems consisting of the differential equation and following boundary value conditions: We give the following assumptions: We will impose growth conditions on f to obtain two positive solutions of BVP (1.3)-(1.4).The main results in [1,3,13,14] are corollaries of our theorems.
This paper is organized as follows.In Section 2, we first introduce some definitions and a fixed-point theorem, which is the generalized form of the Leggett-Williams fixedpoint theorem, founded in Avery and Henderson [6], and then we present our main results.Several corollaries to illustrate the main results are given in Section 3.

Main results.
For convenience, we first introduce some definitions in Banach spaces, such as in [6,9], and a fixed theorem, which is a generalization of the Leggett-Williams fixed point theorem, see Avery and Henderson [6].The main results and their proofs will be presented at the end of this section.Definition 2.1.Let X be a real Banach space; a nonempty closed convex set P ⊂ X is called a cone of X if it satisfies the following conditions: (i) x ∈ P , λ ≥ 0 implies λx ∈ P , (ii) x ∈ P , −x ∈ P implies x = 0.
Every cone P ⊂ X induces an ordering in X, which is given by x ≤ y if and only if y − x ∈ P [6].Definition 2.2.A map ψ : P → [0, +∞) is called a nonnegative, continuous, increasing functional, provided ψ is nonnegative and continuous and satisfies ψ(x) ≤ ψ(y) for all x, y ∈ P with x ≤ y.Definition 2.3.An operator is called completely continuous if it is continuous and maps bounded sets into precompact sets.Denote (2.1) Lemma 2.4 [6].Let X be a real Banach space, P a cone of X, γ and φ two nonnegative increasing continuous maps, θ a nonnegative continuous map with θ(0) = 0. Suppose there are two positive numbers c and M such that x ≤ Mγ(x) for x ∈ P (γ,c). (2.2) Again, assume T : P (γ,c) → P is completely continuous, and that there are positive numbers 0 < a < b < c such that ) > a and P (φ,a) = ∅ for x ∈ ∂P (φ, a).Then T has at least two fixed points x 1 and x 2 ∈ P (γ,c) satisfying (2.4) The following lemma is similar to Lemma 2.4, whose proof is omitted.
Lemma 2.5.Let X be a real Banach space, P a cone of X, γ and φ two nonnegative increasing continuous maps, θ a nonnegative continuous map, and θ(0) = 0. Suppose there are two positive numbers c and M such that x ≤ Mγ(x) for x ∈ P (γ,c). (2.5) Again, assume T : P (γ,c) → P is completely continuous, and that there are positive numbers 0 < a < b < c such that ) < a and P (φ,a) = ∅ for x ∈ ∂P (φ, a).Then T has at least two fixed points x 1 and x 2 ∈ P (γ,c) satisfying To be able to apply Lemmas 2.4 and 2.5, we must define an operator on a cone in a suitable Banach space.In order to do this, we first observe the Green functions for the above nth-order three-point boundary value problem.
has the unique solution is the unique solution of (2.8).One gets

12)
and then Substitute A into (2.11).Then the first part of the lemma is complete.
To prove that u(t . This is simple and is omitted.
∞ . (2.14) We note that, for y ∈ E with y (i) (0) = 0 for i = 0, 1,...,n− 2, Hence, y ∞ ≤ y ∞ and y ∞ ≤ y ∞ .By bootstrapping, one sees that (2.17) Define the subset of E by (2.18) Define an operator T by (2.20) Hence, we get the following lemma.Lemma 2.7.Assume (H 1 ) and (H 2 ).Then (i) P is a cone in Banach space E; (ii) T P ⊂ P and T is completely continuous; 3) and (1.4) if and only if y is a fixed point of the operator T in the P .
Proof.The proofs of (i)-(v) are simple and are omitted.
From now on, fix l such that 0 < η < l < 1, and define the nonnegative, increasing, continuous functionals γ, θ, and φ by for every u ∈ P .We see that γ(u) = θ(u) ≤ φ(u).In addition, for each We also find that Finally, for notational convenience, we denote We now present our first result of this paper.

.25)
Proof.To begin, we define a completely continuous operator T : P → E as above for every u ∈ P .Obviously, From the definition of T and Lemma 2.7, we claim that for each u ∈ P , w = T u ∈ P and satisfies (1.4) and w(1) is the maximum value of w on [0, 1].
It is well known that each fixed point of T in P is a solution of (1.3)-(1.4).We proceed to verify that the conditions of Lemma 2.4 are met.
As a result of Lemma 2.7, we conclude that T : P (γ,c) → P and T is completely continuous.We now show that (i), (ii), (iii) of Lemma 2.4 are satisfied.
Firstly, we prove that Lemma 2.4(i) is satisfied.For each u ∈ ∂P (γ, c), (2.28) As a consequence of (A), (2.30) Secondly, we show that Lemma 2.4(ii) is fulfilled.We choose u ∈ ∂P (θ, b).Then Thus u (i) (t) ≥ 0 for all t ∈ [0, 1], i = 0, 1,...,n− 2, and (2.34) By (B), we have and so (2.36) Finally, we verify that Lemma 2.4(iii) is also satisfied.It is easy to show that P (φ,a) ≠ ∅. (2.37) From assumption (C), we have u (i) (t) ≥ 0 for all t ∈ [0, 1] and i = 0, 1,...,n− 2, and and so (2.39) Therefore, BVP (1.3)-(1.4)has at least two positive solutions u 1 and u 2 in P (γ,c) such that This completes the proof of Theorem 2.8.Now we deal with the following boundary value problem: ∞ . (2.45) It is easy to see that y = y (n−2) ∞ for all y ∈ E. Define the cone P ⊂ E by The method is just similar to what we have done above.We choose a fixed number l ∈ (0,η), and define the nonnegative, increasing functionals γ, θ, and φ on P , respectively, as (2.47) Define the operator T : P → X by By a method similar to that of Theorem 2.8, we have the following theorem and its proof is omitted.
Proof.Firstly, by (ii), choosing b = x 0 η, one gets Secondly, choose K sufficiently large such that Thirdly, choose K 1 sufficiently large such that where λ, ξ, and λ l are given in Theorem 2.9.Then BVP (2.41) has at least two positive solutions.
Proof.The proofs of Theorems 3.3 and 3.4 are similar to that of Theorem 3.1 and are omitted.