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

Li, Hongyu; Zhang, Junting

doi:https://doi.org/10.1155/2018/6486135

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Volume 2018 | Article ID 6486135 | https://doi.org/10.1155/2018/6486135

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

Hongyu Li¹and Junting Zhang¹

Academic Editor: Yong H. Wu

Received04 Jul 2018

Revised07 Nov 2018

Accepted25 Nov 2018

Published11 Dec 2018

Abstract

By using fixed point theorems with lattice structure, the existence of negative solution and sign-changing solution for some second-order multipoint boundary value problems is obtained.

1. Introduction

In this paper, the following second-order ordinary differential equation will be considered: subject to the multipoint boundary condition where , and .

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 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 [28–34]). At present, a few authors have used those fixed point theorems with lattice structure to study boundary value problems (see [6, 17, 28–37]). 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 three-point boundary value problem: where is continuous, . 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, 28–37], 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.

2. 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 such that implies . is called solid if , 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 and exist for arbitrary .

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 [28–31]). 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 , then there exist and , such that For . Let Then is also a cone in .

Definition 2 (see [30, 31, 41]). If there exist , and such that (10) holds, and maps into , then the positive linear operator is said to satisfy condition.
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 condition and is quasi-additive on lattice. Assume that
there exist and such that there exist and such that , 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 . In addition, assume that
, is an even number;

Then the operator has at least one negative fixed point and one sign-changing fixed point.

3. Main Results

For convenience, we list the following conditions.

is continuous,

The sequence of positive solutions of the equation

is

uniformly on .

Let with supremum norm . Set , the is a solid cone in . And under the partial order which is induced by , is a lattice.

In the following, we define some operators , and : where Obviously, , 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 .

Lemma 6. The linear operator satisfies condition.

Proof. By , Lemma 5, and the definition of the spectral radius, we know that By (20), we have By (19) and (23), we have From (24) and (25), we have By (19), we have Hence, by adding (26) to (27), we have i.e., where
Let where . Obviously, By Krein-Rutman theorem, there exist and such that By (29) and (32), we obtain Set Obviously, , and by (34), for , we have That is, From (33) and (35), we have By (37), we have where .
Therefore, from (31), (36), and (38), it is easy to know that the linear operator satisfies condition.

Theorem 7. Suppose that , , and hold. In addition, assume that there exists such thatIf , where is defined by , then the boundary value problem (1), (2) has at least one nontrivial solution.

Proof. By , 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.
For , let and then .
By , we have From (42), we know that is quasi-additive on lattice.
From (39) and (40), there exists such that By (43) and (44), we have Let . Then by (45) and (46), we have i.e., where Obviously, we have In the following, we prove that .
In fact, by , we have . From , , when , we have i.e.,So Therefore, by (52), we have Soi.e.,Since , we know that 1 is not the eigenvalue of by Lemma 6 and .
By the above proof, we know that the conditions of Lemma 3 hold. So by Lemma 3, the boundary value problem (1), (2) has at least one nontrivial solution.

Theorem 8. Assume that , (39), and (40) are satisfied. In addition, suppose that , and , where is a natural number. Then the boundary value problem (1), (2) has at least one negative solution and one sign-changing solution.

Proof. By (17), for , we have From (56) and (57), we obtain that Similarly, we know that Since , we have , and So we have By (58)-(61), we have Let be the sum of algebraic multiplicities for all the eigenvalues of, lying in the interval . By (55), Lemma 5, and , we know that 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.

4. Examples

We consider second-order four-point boundary value problem

By simple calculations, , , , and are solutions of the equation

Example 1. Choose By (66), it is easy to know that is continuous, and By calculation, . We can choose . Then we have So by Theorem 7, the boundary value problem (64) has at least one nontrivial solution.

Example 2. Choose By (68), we know that is continuous, , and . By calculation, . We can choose . Then we have So by Theorem 8, the boundary value problem (64) has at least one negative solution and one sign-changing solution.

Example 3. Choose By (70), we know that is continuous, , and . By calculation, . We can choose . Then we have So by Theorem 8, the boundary value problem (64) has at least one negative solution and one sign-changing solution.

Data Availability

The date underlying the findings of our manuscript can be obtained by simple calculations. We did not quote other data.

Conflicts of Interest

The authors declare that they have no conflicts of interest regarding the publication of this paper.

Acknowledgments

The project is supported by the Youth Science Foundation of China (11801322), the National Natural Science Foundation of China (11571207), and Shandong Natural Science Foundation (ZR2018MA011).

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Copyright © 2018 Hongyu Li and Junting Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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