Multiple

Using a fixed point theorem in ordered Banach spaces with lattice structure founded by Liu and Sun, this paper investigates the multiplicity of nontrivial solutions for fourth order m-point boundary value problems with sign-changing nonlinearity. Our results are new and improve on those in the literature.


Introduction
Consider the following fourth order differential equation x (4) (t) = f (t, x(t)), t ∈ (0, 1) (1.1) subject to one of the following two classes of m-point boundary value conditions and x ′ (0) = 0, x(1) = m−2 i=1 α i x(η i ); x ′′′ (0) = 0, x ′′ (1) = m−2 i=1 α i x ′′ (η i ), (1.3) where f : R × R → R is a given sign-changing continuous function, m ≥ 3, η i ∈ (0, 1) and The existence of nontrivial or positive solutions of nonlinear multi-point boundary value problems (BVP, for short) for fourth order differential equations has been extensively studied and lots of excellent results have been established by using fixed point index for cone mappings, standard upper and lower solution arguments, fixed point theorems for cone mappings and so on (see [2,[8][9][10] and the references therein).For example, in [9], Wei and Pang studied the following fourth order differential equation with the boundary condition (1.2).
By means of fixed point index theory in a cone and the Leray-Schauder degree, the existence and multiplicity of nontrivial solutions are obtained.
Recently Professor Jingxian Sun advanced a new approach to compute the topological degree when the concerned operators are not cone mappings in ordered Banach spaces with lattice structure.He established some interesting results for such nonlinear operators (for details, see [3,6,7]).To our best knowledge, there is no paper to use this new method to study fourth order m-point boundary value problems.We try to fill this gap in the present paper.
Suppose the following conditions are satisfied throughout.
(H0) the sequence of positive solutions of the equation This paper is organized as follows.In Section 2, we present some basic definitions of the lattice and some lemmas that will be used to prove the main results.In Section 3, we shall give our main results and their proofs.
Let E be an ordered Banach space in which the partial ordering ≤ is induced by a cone P ⊆ E. P is called normal if there exists a constant N > 0 such that θ ≤ x ≤ y implies x ≤ N y .
Definition 2.1. [7]We call E a lattice under the partial ordering ≤, if sup{x, y} and inf{x, y} exist for arbitrary x, y ∈ E.
Definition 2.2. [3]Let E be a Banach space with a cone P and A : E → E be a nonlinear operator.We say that A is a unilaterally asymptotically linear operator along P w = {x ∈ E : x ≥ w, w ∈ E}, if there exists a bounded linear operator L such that L is said to be the derived operator of A along P w and will be denoted by A ′ Pw .Similarly, we can also define a unilaterally asymptotically linear operator along A is a unilaterally asymptotically linear operator along P and (−P ).It is remarkable that A is not assumed to be a cone mapping.Definition 2.3. [7]Let D ⊆ E and A : D → E be a nonlinear operator.A is said to be quasi-additive on a lattice, if there exists v * ∈ E such that The following lemma is important for us to obtain the main results.
Lemma 2.1. [3]Suppose E is an ordered Banach space with a lattice structure, P is a normal cone of E, and the nonlinear operator A is quasi-additive on the lattice.Assume that (i) A is strongly increasing on P and (−P ); (ii) both A ′ P and A ′ (−P ) exist with r(A ′ P ) > 1 and r(A ′ −P ) > 1, and 1 is not an eigenvalue of A ′ P and A ′ (−P ) corresponding a positive eigenvector; Then A has at least three nontrivial fixed points containing one sign-changing fixed point.
Then E is a Banach space and P is a normal cone of E. It is easy to see that E is a lattice under the partial ordering ≤ that is deduced by P .
Using the same method as in [9], we can easily convert BVP (1.1) and (1.2) into the following operator equation EJQTDE, 2010 No. 55, p. 3 where the operators F and L 1 are defined by where Similarly, we can convert BVP (1.1) and (1.3) into the following operator equation where Define then the following lemma is obvious.

Lemma 2.2. x(t) is a solution of the BVP (1.1) and (1.2) (BVP (1.1) and (1.3)) if and only if x(t) is a solution of the operator equation
) is quasi-additive on the lattice; (iv) the sequences of all eigenvalues of the operators L , where r(L i ) is the spectral radius of the operator L i (i = 1, 2).EJQTDE, 2010 No. 55, p. 4 Proof.The proof of (i)-(iii) is obvious.Now we start to prove the conclusions (iv) and (v).
Let µ be a positive eigenvalue of the linear operator L 2 1 , and y ∈ E \ {θ} be an eigenfunction corresponding to the eigenvalue µ.Then we have By properties of differential operators, if (2.7) has a nonzero solution, then there exists Substituting this solution into (2.7),we have and the eigenfunction corresponding to the eigenvalue 1 where C is a nonzero constant.By the ordinary method, we can show that any two eigenfunctions corresponding to the same eigenvalue Now we show that (2.9) Obviously, we need only show that For any y ∈ ker( 1 )y is an eigenfunction of the linear operator L 2 1 corresponding to the eigenvalue 1 Then there exists a nonzero constant γ such that EJQTDE, 2010 No. 55, p. 5 By direct computation, we have (2.11) It is easy to see that the general solution of (2.11) is of the form where C 1 , C 2 , C 3 , C 4 are nonzero constants.
By the Schwarz inequality, we obtain EJQTDE, 2010 No. 55, p. 6 Applying the condition sin which is a contradiction to m−2 i=1 α i < 1.Thus, (2.9) holds.It follows from (2.8) and (2.9) that the algebraic multiplicity of the eigenvalue 1 Similarly, we can show that the sequence of all eigenvalues of the operator .
Suppose that f satisfies (H1).Then there exists R > 0 such that for a given ε > 0,

and the algebraic multiplicity of 1 s 2 n is 1 .
By the definition of the spectral radius, we have