On the Nonhomogeneous Fourth-Order p-Laplacian Generalized Sturm-Liouville Nonlocal Boundary Value Problems

We study the nonlinear nonhomogeneous -point generalized Sturm-Liouville fourth-order -Laplacian boundary value problem by using Leray-Schauder nonlinear alternative and Leggett-Williams fixed-point theorem.

Recently, much attention has been paid to the existence of positive solutions for nonlocal nonlinear boundary value problems BVPs for short , see 1-4 and references therein.Such problems have potential applications in physics, biology, chemistry, and so forth.For example, a second-order three-point is used as a model for the membrane response of a spherical cap in nonlinear diffusion generated by nonlinear sources and in chemical reactor theory.
At the same time, the boundary value problems with p-Laplacian operator have been discussed extensively, for example, see 1-3, 5-7 .
In 1 , Feng et al. researched the boundary value problem they obtained at least one or two positive solutions under some assumptions imposed on the nonlinearity of f by applying Krasnoselskii fixed-point theorem.
Zhou and Ma studied the existence and iteration of positive solutions for the following third-order generalized right-focal boundary value problem with p-Laplacian operator in 3 : they established a corresponding iterative scheme for 1.4 by employing the monotone iterative technique.We would also like to mention the work of Zhang and Liu in 7 , in which they considered the existence of positive solutions for by virtue of monotone iterative techniques, and they established a necessary and sufficient condition of positive solutions for their problem.However, to the best of our knowledge, there are not many results concerning about the existence and multiple solutions of fourth-order p-Laplacian generalized Sturm-Liouville n-point boundary value problems.In this paper, motivated by the study of 4, 8 , we committed to consider the fourth-order p-Laplacian generalized Sturm-Liouville nonlocal boundary value problem without assuming any monotonicity condition on the nonlinearity f.
The rest of the paper is arranged as follows.We state some definitions and several preliminary results in Section 2 that we will use in the sequel.Then in Section 3 we present the existence of one positive solution of BVP 1.1 by Leray-Schauder nonlinear alternative.In Section 4 we get three solutions by Leggett-Williams fixed-point theorem.

Preliminaries and Some Lemmas
The basic space used in this paper is E C 0, 1 .It is well known that E is a real Banach space with the norm u max t∈ 0,1 |u t |.Denote

2.1
Definition 2.1.A function u is said to be a solution of the boundary value problem 1.1 if u ∈ C 2 0, 1 satisfies 1.1 and φ p u ∈ C 2 0, 1 .In addition, u is said to be a positive solution if u t > 0 for t ∈ 0, 1 , and u is a solution of BVP 1.1 .
Throughout the paper, we assume the following condition is satisfied:

Let y t
−φ p u t , then BVP 1.1 is divided into the following two parts:

2.3
It is not difficult that we can transform 2.2 into the following differential equations:

2.4
By routine calculations we can get the following three Lemmas.
Lemma 2.2.The BVP 2.4 has a unique solution where

2.6
Lemma 2.3.The BVP 2.3 has a unique solution where G 2 ξ i , s φ q y s ds .

2.8
The proof of Lemma 2.3 is similar to that of Lemma 5.5.1 in 8 , so we omit it here.From Lemmas 2.2 and 2.3 we can get that u t is a solution of BVP 1.1 if and only if

2.16
Thus we can get that min t∈ ,1− Tu t λ Tu , which means T K ⊆ K.
We present here several definitions.Given a cone K in a real Banach space E, a map α is said to be a nonnegative continuous concave resp., convex functional on K provided that α : K → 0, ∞ is continuous and 2.17 for all, x, y ∈ K and t ∈ 0, 1 .Let 0 < a < b be given, and let α be a nonnegative continuous concave functional on K. Define the convex sets P r and P α, a, b by

2.18
For the convenience of the reader, we present here the Leggett-Williams fixed-point theorem and the Leray-Schauder nonlinear alternative theorem.Then A has at least three fixed points x 1 , x 2 , and x 3 and such that x 1 < a, b < α x 2 and x 3 > a, with α x 3 < b.
Now we cite the Leray-Schauder nonlinear alternative.
Lemma 2.7 see 10 .Let F be a Banach space and Ω a bounded open subset of F, 0 ∈ Ω. T : Ω → F be a completely continuous operator.Then, either there exists x ∈ ∂Ω, λ > 1 such that T x λx, or there exists a fixed point x * ∈ Ω.

Results of One Nontrivial Solution
In this section, we study the existence of one nontrivial solution of BVP 1.1 by Leray-Schauder nonlinear alternative. Denote

3.3
In the same way, we obtain

3.4
Thus we have

Results of Multiple Positive Solutions
In the following parts, we will study the existence of multiple positive solutions of BVP 1.1 by using Leggett-Williams fixed-point theorem. Denote Define the nonnegative continuous concave functional on K by α u min and A, B, Λ, Λ 0 , Λ 1 be defined in Sections 2 and 3. We list the following three hypotheses: Proof.Firstly, we prove that T : P c → P c .The operator T is completely continuous.
From condition H1 , we can get

4.4
Thus we get Tu c; therefore, T : P c → P c .The operator T is completely continuous by an application of Ascoli-Arzela theorem.
In the same way, condition H2 implies that condition A2 of Lemma 2.6 is satisfied.In the following, we show that condition A1 of Lemma 2.6 is satisfied.

4.8
Therefore, condition A1 of Lemma 2.6 is satisfied.Finally, we show that condition A3 of Lemma 2.6 is satisfied.
If u ∈ P α, b, c , and Tu > b/λ, then α Tu t min t 1− Tu t λ Tu > b.Therefore, condition A3 of Lemma 2.6 is also satisfied.By Lemma 2.6, there exist three positive solutions u 1 , u 2 , and u 3 such that u 1 < a, b < min t∈ ,1− u 2 t , and u 3 > a, with min t∈ ,1− u 3 t < b.Thus we completed the proof.

Lemma 2 . 6
see 9 , Leggett-Williams fixed-point theorem .Let A : P c → P c be a completely continuous operator, and let α be a nonnegative continuous concave functional on K such that α x x for all x ∈ P c .Suppose there exist 0 < a < b < d c such that A1 {x ∈ P α, b, d : α x > b} / ∅, and α Ax > b for x ∈ P α, b, d ; A2 Ax < a for x a; A3 α Ax > b for x ∈ P α, b, c with Ax > d.