Existence of Nontrivial Solutions for Some Second-Order Multipoint Boundary Value Problems

where βi > 0, i = 1, 2, ⋅ ⋅ ⋅ , m − 2; ∑m−2 i=1 βi < 1; 0 < α1 < α2 < ⋅ ⋅ ⋅ < αm−2 < 1, and ∑m−2 i=1 βiαi < 1. The multipoint boundary value problems of ordinary differential equations arise in different areas of applied mathematics and physics. In 1992, Gupta studied nonlinear second-order three-point boundary value problems (see [1]). Since then, different types of nonlinear multipoint boundary value problems have been studied. Up to now, many great achievements about multipoint boundary value problems have been made. For example, many authors have investigated the existence of nontrivial solutions for nonlinear multipoint boundary value problems. Most of them have used upper and lower solution method, fixed point index theory, Guo-Krasnosel’skii fixed point theorem, bifurcation theory, fixed point theorems on cones, and so on (see [2– 27] and references therein). For instance, in [2], the author considered the second-order multipoint boundary value problem y󸀠󸀠 (t) + f (y) = 0, 0 ≤ t ≤ 1, y (0) = 0, y (1) = m−2 ∑

(3) By using fixed point index and Leray-Schauder degree methods, the author showed existence of multiple sign-changing solutions for the boundary value problem (3).In [14], the authors have considered the following multipoint boundary value problem: The authors have used global bifurcation method to obtain the existence of positive solution of the boundary value problem (4).
In recent years, some authors combine the theory of lattice and the theory of topological degree, so they have obtained some fixed point theorems with lattice structure for nonlinear operators which are not assumed to be cone mappings (see [28][29][30][31][32][33][34]).At present, a few authors have used those fixed point theorems with lattice structure to study boundary value problems (see [6,17,[28][29][30][31][32][33][34][35][36][37]).For example, in [35], by using fixed point theorems with lattice structure, the authors considered the existence of positive solution and sign-changing solution for integral boundary value problem under sublinear condition.In [37], the authors considered the existence of positive solution for fourth-order differential equation with fixed point theorems with lattice structure.In [6], the author considered the following second-order threepoint boundary value problem: where The author used fixed point theorems with lattice structure to study the existence of sign-changing solutions for the boundary value problem (5) under the unilaterally asymptotically linear condition.
Motivated by [6,17,[28][29][30][31][32][33][34][35][36][37], we shall study the existence of nontrivial solutions for the boundary value problem (1), (2).In this paper, we assume that the nonlinear term satisfies superlinear conditions concerning the first eigenvalue corresponding to the relevant linear operator.The method we use is fixed point theorems with lattice structure.And we obtain the sufficient condition about the existence of negative solution and sign-changing solution for the boundary value problem (1), (2).The method is different from those of [2,4].And the main results are different from those of the work [2,4].This paper is arranged as follows.In Section 2, we give some definitions and fixed point theorems with lattice structure.In Section 3, we shall give some lemmas and the main results about the existence of nontrivial solutions (including negative solution and sign-changing solution) for the boundary value problem (1), (2).Finally, in Section 4, some examples are given to illustrate our main results.

Preliminaries
Let  be an ordered Banach space in which the partial ordering ≤ is induced by a cone  ⊂ . is called normal if there exists a positive constant  > 0 such that  ≤  ≤ V implies ‖‖ ≤ ‖V‖. is called solid if int P ̸ = , i.e.,  has nonempty interior. is called total if  =  − .If  is solid, then  is total.For the concepts and the properties about the cones, we refer to [31,38,39].
We call  a lattice under the partial ordering ≤, if sup{, V} and inf{, V} exist for arbitrary , V ∈ .
For  ∈ , let + and  − are called positive part and negative part of , respectively.Taking || =  + +  − , then || ∈ .For the definition and the properties of the lattice, we refer to [40].
For convenience, we use the following notations: and clearly Definition 1 (see [28][29][30][31]).Let  ⊂  and  :  →  be a nonlinear operator.If there exists  * ∈  such that then  is said to be quasi-additive on lattice.
Let  :  →  be a bounded linear operator.If () ⊂ , then the operator  is called to be positive.
In this section, we assume that  is a Banach space,  is a total cone, the partial ordering ≤ in  is induced by , and  is a lattice in the partial ordering ≤.
Let  :  →  be a positive completely continuous linear operator;  * the conjugated operator of ; () a spectral radius of ; and  * the conjugated cone of .Since  ⊂  is a total cone, by Krein-Rutman theorem, we can infer that if () ̸ = 0, then there exist  ∈  \ {} and  * ∈  * \ {}, such that For  > 0. Let Then ( * , ) is also a cone in .
Let  be a cone of a Banach space .If  ∈ ( \ {}) is a fixed point of , then  is said to be a positive fixed point of .If  ∈ ((−) \ {}) is a fixed point of operator , then  is said to be a positive fixed point of operator .If  ∈ ( \ {}) is a fixed point of operator , then  is said to be a negative fixed point of operator .If  ∉ ( ∪ (−)) is a fixed point of operator , then  is said to be a sign-changing fixed point of operator .
In [30], Sun and Liu considered computation for the topological degree about superlinear operators which are not cone mappings and obtained the following results.Lemma 3. Let the cone  ⊂  be solid, and  :  →  be a completely continuous operator, and  = , where  is a positive completely continuous linear operator satisfying H condition and  is quasi-additive on lattice.Assume that (i) there exist  1 >  −1 () and (ii) there exist 0 <  2 <  −1 () and  2 ∈  such that (iii)  = , the Fréchet derivative    of  at  exists, and 1 is not an eigenvalue of   .Then the operator  has at least one nonzero fixed point.In [31], Sun further obtained the following result about the existence of sign-changing fixed points for superlinear operators.
Lemma 4. Let the conditions in Lemma 3 hold, and  denote the sum of the algebraic multiplicities for all eigenvalues of    lying in (1, +∞).In addition, assume that (iv)  ̸ = 0,  is an even number; Then the operator  has at least one negative fixed point and one sign-changing fixed point.

Main Results
For convenience, we list the following conditions.
The sequence of positive solutions of the equation is And under the partial order ≤ which is induced by ,  is a lattice.
In the following, we define some operators , , and Φ: where Obviously,  = Φ, and the nontrivial fixed points of the operator  are nontrivial solutions of the boundary value problem (1), (2) (see [3]).
Lemma 5 (see [2]).Let  be a positive number, and the linear operator  be defined by (17).Eigenvalues of the linear operator  are and algebraic multiplicity of /  is equal to 1, where   is defined by (C 2 ).
Lemma 6.The linear operator  satisfies H condition.
Proof.By (C 1 ), we easily know that  :  →  is a completely continuous operator, and  :  →  is a bounded positive linear completely continuous operator (see [3]).By Lemma 6, we know that the linear operator  satisfies H condition.
(63) By ( 62) and (63), we know that the conditions (iv) and (v) in Lemma 4 hold.By the proof of Theorem 7, the conditions (i), (ii), and (iii) in Lemma 4 are satisfied.Therefore, by Lemma 4, the boundary value problem (1), ( 2) has at least one negative solution and one sign-changing solution.

Examples
We consider second-order four-point boundary value problem ) . (
8, the boundary value problem (64) has at least one negative solution and one sign-changing solution.