Positive Solutions for Singular Complementary Lidstone Boundary Value Problems

and Applied Analysis 3 i ‖Au‖ ≤ ‖u‖, u ∈ P ∩ ∂Ω1 and ‖Au‖ ≥ ‖u‖, u ∈ P ∩ ∂Ω2 or ii ‖Au‖ ≥ ‖u‖, u ∈ P ∩ ∂Ω1 and ‖Au‖ ≤ ‖u‖, u ∈ P ∩ ∂Ω2. Then A has a fixed point in P ∩ Ω2 \Ω1 . This paper is organized as follows: in Section 2, some preliminaries are given; in Section 3, we give the existence results. 2. Preliminaries First, it is clear to see that the boundary value problem 1.1 , −1 u 2n 1 t h t f u t , in 0 < t < 1, u 0 0, u 2i 1 0 u 2i 1 1 0, 0 ≤ i ≤ n − 1, 2.1 is equivalent to the system u′ t v, in 0 < t < 1, −1 v 2n t h t f u t , in 0 < t < 1, u 0 0, v 2i 0 v 2i 1 0, 0 ≤ i ≤ n − 1. 2.2 Next, Problem 2.2 can be easily transformed into a nonlinear 2n-order ordinary differential equation. Briefly, the initial value problem, u′ t v, in 0 < t < 1, u 0 0, 2.3


Introduction
In this paper, we are concerned with the existence of positive solutions for the following nonlinear differential equation: where n ∈ N, h t ∈ C 0, 1 , 0, ∞ , and h t may be singular at t 0 or t 1; f ∈ C 0, ∞ , 0, ∞ .
Recently, on the boundary value problems of 2nth-order ordinary differential equation system −1 n u 2n t λh t f u t , 1.2 many authors have established the existence and multiplicity of positive solutions of 1.2 by means of the method of upper and lower solutions and fixed point theorem, see 1-7 and references therein.More recently, the complementary Lidstone problem: −1 n u 2n 1 t h t, u t , . . ., u q t , n ≥ 1, q fixed, 0 ≤ q ≤ 2n in 0 < t < 1, u 0 a 0 , u 2i 1 0 a i , u 2i 1 1 was discussed in 8 .Here, h : 0, 1 × R q 1 → R is continuous at least in the interior of the domain of interest.Existence and uniqueness criteria for the above problem are proved by the complementary Lidstone interpolating polynomial of degree 2n.In 9 , the authors have studied the existence of positive solutions of singular complementary Lidstone problems on the basis of a fixed-point theorem of cone compression type.As far as we know, no papers are concerned with the multiplicity of positive solutions for 1.1 .Therefore, inspired by the above references, we will show the existence and multiplicity of positive solutions of 1.1 .The proof of our results is based on the following fixed-point theorems in a cone Let E be a real Banach space with norm • and P ⊂ E a cone in E, P r {x ∈ P : x < r} r > 0 .Then P r {x ∈ P : x ≤ r}.A map α is said to be a nonnegative continuous concave functional on P if α : P → 0, ∞ is continuous and for all x, y ∈ P and t ∈ 0, 1 .For numbers a, b such that 0 < a < b and α is a nonnegative continuous concave functional on P , we define the convex set P α, a, b {x ∈ P : a ≤ α x , x ≤ b}.iii α Ax > a for x ∈ P α, a, c with Ax > b.
Then A has at least three fixed points x 1 , x 2 , x 3 satisfying Then A has a fixed point in This paper is organized as follows: in Section 2, some preliminaries are given; in Section 3, we give the existence results.

Preliminaries
First, it is clear to see that the boundary value problem 1.1 ,

2.2
Next, Problem 2.2 can be easily transformed into a nonlinear 2n-order ordinary differential equation.Briefly, the initial value problem, can be solved as Then, inserting 2.4 into the second equation of 1.1 , we have

2.5
Finally, we only need to consider the existence of positive solutions of 2.5 .The function Let G n t, s be the Greens function of the following problem:

2.6
By induction, the Greens function G n t, s can be expressed as see 2 where

2.8
So it is easy to see that where Therefore, the solution of 2.5 can be expressed as

2.12
We now define a mapping

3.2
Then 2.5 or 1.1 has at least one positive solution.

3.5
On the other hand, since lim r → ∞ f r /r ∞, there exists R > 0 such that f r r ≥ 4 σθ n 1/4 Theorem 3.2.Assume (H1) holds.In addition, suppose that the following conditions hold:

3.11
Then 2.5 or 1.1 has at least one positive solution.

3.14
On the other hand, since lim r → ∞ f r /r 0, there exists R > 0 such that , for r ≥ R.

3.16
Consequently, 3.17 Therefore, by Lemma 1.2, 1.1 has at least one positive solution.
Theorem 3.3.Assume that (H1) holds.In addition, the function f is nondecreasing and satisfies the following growth conditions: Proof.For the sake of applying the Leggett-Williams fixed-point theorem, define a functional α u on cone P by v t , ∀v ∈ P.

3.21
Evidently, α : P → R is a nonnegative continuous and concave.Moreover, α v ≤ v for each v ∈ P .Now we verify that the assumption of Lemma 1.1 is satisfied.Firstly, it can verify that there exists a positive number c with c ≥ b a/σ such that T : By H4 , it is easy to see that there exists τ > 0 such that f r r < 1 2 , ∀r ≥ τ.

3.23
If v ∈ P c , then 3.24 by H1 and H3 .

3.25
Then for each v ∈ P d , we have

3.26
Finally, we will show that {v ∈ P α, a, b : α v > a} / ∅ and α Tv > a for all v ∈ P α, a, b .

3.30
Above all, we know that the conditions of Lemma 1.1 are satisfied.By Lemma 1.1, the operator T has at least three fixed points v i i 1, 2, 3 such that

3.31
The proof is complete.
Example 3.4.If n 1, then consider the boundary value problem:

3.32
Example 3.5.If n 2, then consider the boundary value problem:

3.33
It is obvious to see that Examples 3.4 and 3.5 satisfy the assumptions of Theorems 3.1 and 3.2.

3.35
It is obvious that f is continuous and H1 holds.On the other hand, since u 2 / 2 u u 2 4u / 2 u 2 ≥ 0, for 0 ≤ u, it is clear to see that 195u 2 / 2 u is nondecreasing for 0 ≤ u < 1, and u − 1 1/2 65 is also nondecreasing for u ≥ 1.In addition,