EXISTENCE OF THREE SOLUTIONS FOR SYSTEMS OF MULTI–POINT BOUNDARY VALUE PROBLEMS

By using a variational approach, we obtain some sufficient conditions for the existence of three classical solutions of a boundary value problem consisting of a system of differential equations and some multi–point boundary conditions. Applications of our results are discussed. Our results extend some related work in the literature.

Then there exist an open interval Λ ⊆ [0, ∞) and a positive real number δ such that, for each λ ∈ Λ, BVP (3), (4) has at least three solutions belonging to ]) are less than δ.
For the reader's convenience, we now recall the following two results that are fundamental tools in our discussion.
Lemma 1.1 ([25, Proposition 3.1]) Let X be a separable and reflexive real Banach space, and Φ, J two real functions on X. Assume that there exist r > 0 and u 0 , u 1 ∈ X such that Then, for each η satisfying Lemma has at least three solutions in X whose norms are less than δ.
The rest of this paper is organized as follows.After this introduction, in Section 2, we state the main results and give one simple example for illustrative purposes.The proofs of the main results, together with some technical lemmas, are presented in Section 3.

Main Results
In the sequel, for i = 1, . . ., n, let and let X be defined by where Then, X is a separable and reflexive real Banach space.
We first make the following assumption. (H1) We now introduce some notations.For any nonempty set S, let and for x > 0, define EJQTDE Spec.Ed.I, 2009 No. 10 Let the positive constants κ i , i = 1, . . ., n, and ρ be defined by and Throughout this paper, the following assumptions are also needed.
(H2) there exists a function (H3) there exist two positive constants c and d with c < d such that and where (H4) there exist θ ∈ L 1 (0, 1) and n positive constants γ i with γ i < p i , i = 1, . . ., n, such that We say that a function u = (u 1 , . . ., u n ) ∈ X is a weak solution of BVP (1), (2) if ).Under the assumptions (H1)-(H5), Theorem 2.1 below shows that BVP (1), (2) has at least three classical solutions.To prove the theorem, we will first apply Lemmas 1.1 and 1.2 to obtain the existence of three weak solutions of BVP (1), ( 2), then we show that the three weak solutions are indeed the classical solutions.In the process of the proof, three functionals Φ, Ψ, and J are constructed in such a way that all the conditions of Lemmas 1.1 and 1.2 are satisfied.
We now make some brief comments about the assumptions (H1)-(H5).( H1) is needed to obtain some useful bounds for functions in X (see Lemma 3.1).The function F introduced in (H2) is used in the construction of the functional Ψ. (H3) is required in the proof of the existence of a function w ∈ X with some nice properties (see Lemma 3.2).Both Lemmas 3.1 and 3.2 are crucial to prove our existence results.(H4), together with Lemma 3.1, is used to show the functional Φ(u) + λΨ(u) is weakly coercive for λ ∈ [0, ∞), i.e., lim Finally, (H5) is needed to show that J(0, . . ., 0) = 0, which is necessary in order to apply Lemma 1.1.Now, we state our main results.The first one is concerned with BVP (1), (2).For n = 1, let The following corollaries are direct consequences of Theorem 2.1.
Then there exist an open interval Λ ⊆ [0, ∞) and a positive real number δ such that, for each λ ∈ Λ, BVP (5), (7) has at least three classical solutions whose norms in X are less than δ.
Corollary 2.2 Assume (H1) and the following conditions hold: where ρ is defined by (11) and κ is defined by (13), and G(t) = t 0 g(s)ds.(B3) there exists σ > 0 and γ > 0 with γ < p such that Then there exist an open interval Λ ⊆ [0, ∞) and a positive real number δ such that, for each λ ∈ Λ, BVP (5), (7) has at least three classical solutions whose norms in X are less than δ.

Corollary 2.3 Assume (A2) and the following condition hold:
(C1) there exist two positive constants c and d with c < d such that Then there exist an open interval Λ ⊆ [0, ∞) and a positive real number δ such that, for each λ ∈ Λ, BVP (5), (4) has at least three classical solutions whose norms in X are less than δ.
In particular, for the case when p = 2, by taking c = √ 2c and d = d, it is easy to see that Proposition 1.1 is a special case of Corollary 2.3.

Z. Du & L. Kong
We conclude this section with the following simple example.
We claim that there exist an open interval Λ ⊆ [0, ∞) and a positive real number δ such that, for each λ ∈ Λ, BVP ( 14), (15) has at least three classical solutions whose norms in X are less than δ.
Proof.From ( 19), we see that lim Hence, the existence and uniqueness of a solution of Eq. ( 20) follow from the fact that α(•; u) is continuous and increasing on R.
Lemma 3.4 The function u(t) = (u 1 (t), . . ., u n (t)) is a solution of BVP (1), (2) if and only if u i (t), i = 1, . . ., n, is a solution of the system of the integral equations where x u,i is the unique solution of Eq. (20).
Proof.This can be verified by direct computations.
(15)he best of our knowledge, no known criteria in the literature can be applied to BVP(14),(15)to obtain the same conclusion as what we get here.