# Sign-changing solution for a third-order boundary-value problem in ordered Banach space with lattice structure

- Xiuli Lin
^{1}Email author and - Zengqin Zhao
^{1}

**2014**:132

https://doi.org/10.1186/1687-2770-2014-132

© Lin and Zhao; licensee Springer. 2014

**Received: **11 December 2013

**Accepted: **7 April 2014

**Published: **22 May 2014

## Abstract

In this paper, the sign-changing solution of a third-order two-point boundary-value problem is considered. By calculating the eigenvalues and the algebraic multiplicity of the linear problem and using a new fixed point theorem in an ordered Banach space with lattice structure, we give some conditions to guarantee the existence for a sign-changing solution.

### Keywords

third-order boundary-value problem sign-changing solution Green’s function fixed point lattice## 1 Introduction

where $f\in C([0,1]\times R,R)$.

where $f\in C(R,R)$, $a\in L(0,1)$ is nonnegative with ${\int}_{0}^{1}{a}^{2}(s)\phantom{\rule{0.2em}{0ex}}ds<1$. For the integral boundary-value problem (1.2), Li and Liu [5] also obtained the existence and multiplicity of the sign-changing solutions in ordered Banach space with the lattice structure.

For the third-order boundary-value problem, the existence and multiplicity of solutions have also been discussed in many papers (see [6–11] and the references therein). However, the research on the sign-changing solutions has been proceeded slowly. For the problem (1.1), Yao and Feng [10, 11] established several existence results for the solutions including the positive solutions using the lower and upper solutions and a maximum principle, respectively. To our knowledge, however, there are fewer papers considered the sign-changing solutions of the problem (1.1). Motivated by the work mentioned above, using the eigenvalues of linear operator, we give an existence result for the sign-changing solutions of the problem (1.1).

The main contribution of this paper are as follows: (a) for the sign-changing solutions of the problem (1.1), to our knowledge, there is no result using the eigenvalues of the linear operator until now; (b) we obtain the eigenvalues and the algebraic multiplicity of the linear problem corresponding the problem (1.1), which is one of the key points that we can use to prove our main result; (c) some conditions are given to guarantee the existence for a sign-changing solution of the problem (1.1).

## 2 Notations and preliminaries

The following results will be used throughout the paper.

Let $E=C[0,1]$, $\parallel u\parallel ={max}_{t\in [0,1]}|u(t)|$. Then *E* is a real Banach space with the norm $\parallel \cdot \parallel $. Let $P=\{u\in E:u(t)\ge 0,t\in [0,1]\}$, and *P* is a normal solid cone of *E*, $\{u\in E:u(t)>0,t\in (0,1]\}\subset \stackrel{\circ}{P}=\{x\in P|x\text{is an interior point of}P\}$.

*K*,

*F*,

*A*be defined by

**Remark 1**(1) $A,K:E\to E$ are completely continuous. (2) $G(t,s)\ge 0$, $t,s\in [0,1]$. In fact, since it is obvious in the other case, we only need to prove the case $0\le s\le t\le 1$. Now we suppose that $0\le s\le t\le 1$. Then

**Definition 2.1** [12]

We call *E* a lattice under the partial ordering ≤, if $sup\{x,y\}$ and $inf\{x,y\}$ exist for arbitrary $x,y\in E$.

**Remark 2** $E=C[0,1]$ is a lattice under the partial ordering ≤ that is deduced by the cone $P=\{u\in E:u(t)\ge 0,t\in [0,1]\}$ of *E*.

**Definition 2.2** [12]

*E*be a Banach space with a cone $P,A:E\u27f6E$ be a nonlinear operator. We call that

*A*is a unilaterally asymptotically linear operator along ${P}_{w}=\{x\in E:x\ge w,w\in E\}$, 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 ${A}_{{P}_{w}}^{\prime}$. Similarly, we can also define a unilaterally asymptotically linear operator along ${P}^{w}=\{x\in E:x\le w,w\in E\}$. Specially, if $w=\theta $, We call that *A* is a unilaterally asymptotically linear operator along *P* and −*P*.

**Definition 2.3** [12]

*A*is said to be quasi-additive on lattice, if there exists ${v}^{\ast}\in E$ such that

where ${x}_{+}={x}^{+}=sup\{x,\theta \}$, ${x}_{-}={x}^{-}=-sup\{-x,\theta \}$.

**Remark 3** It is easy to see that the operators *F* and $A=KF$ defined by (2.1) are both quasi-additive on the lattice $E=C[0,1]$.

Let us list some conditions and preliminary lemmas to be used in this paper.

(H_{1}) $f\in C([0,1]\times R,R)$ is strictly increasing in *u*, and $f(t,u)u>0$ for all $t\in [0,1]$, $u\in R\mathrm{\setminus}\{0\}$.

_{2}) ${lim}_{|u|\to +\mathrm{\infty}}\frac{f(t,u)}{u}={\beta}_{1}$ uniformly on $[0,1]$. There exists a positive integer ${n}_{1}$ such that

(H_{3}) ${lim}_{u\to 0}\frac{f(t,u)}{u}={\beta}_{0}$ uniformly on $[0,1]$, and $0<{\beta}_{0}<{\lambda}_{1}^{3}$.

**Lemma 2.1**

*For any*$f\in C[0,1]$, $u\in {C}^{3}[0,1]$

*is a solution of the following problem*:

*if and only if*$u(t)$

*is a solution of the integral equation*

*where* $G(t,s)$ *is defined by* (2.2).

*Proof*On the one hand, integrating the equation

Moreover, we get $u(0)={u}^{\prime}(0)={u}^{\prime}(1)=0$. □

**Remark 4** Considering Lemma 2.1, we find that *u* is a solution of the problem (1.1) if and only if *u* is a fixed point of the operator $A=KF$.

From the following lemma, we can obtain the eigenvalues and the algebraic multiplicity of the linear operator *K*.

**Lemma 2.2**

*The eigenvalues of the linear operator*

*K*

*are*

*and the algebraic multiplicity of each positive eigenvalue* $\frac{1}{{\lambda}_{n}^{3}}$ *is equal to* 1, *where* $0<{\lambda}_{1}<{\lambda}_{2}<\cdots <{\lambda}_{n}<\cdots $ *are the positive solutions of* (2.3).

*Proof*Let $\overline{\lambda}$ be a positive eigenvalue of the linear operator

*K*, and $y\in E\setminus \{\theta \}$ be an eigenfunction corresponding to eigenvalue $\overline{\lambda}$. By Lemma 2.1, we have

*μ*, $\frac{1}{2}\mu (1+\sqrt{3}i)$, $\frac{1}{2}\mu (1-\sqrt{3}i)$, where $\mu =\frac{1}{\sqrt[3]{\overline{\lambda}}}$. Thus the general solution of (2.4) is

Applying the condition $y(0)={y}^{\prime}(0)=0$, we obtain ${C}_{2}=-{C}_{1}$, ${C}_{3}=\sqrt{3}{C}_{1}$, where ${C}_{1}\ne 0$.

*μ*is one of ${\lambda}_{1},{\lambda}_{2},\dots ,{\lambda}_{n},\dots $ , that is,

*K*and the eigenfunction corresponding to the eigenvalue $\frac{1}{{\lambda}_{n}^{3}}$ is

where *C* is a nonzero constant.

*K*corresponding to the eigenvalue $\frac{1}{{\lambda}_{n}^{3}}$ if $(I-{\lambda}_{n}^{3}K)y\ne \theta $. Considering (2.6), there exists a nonzero constant

*σ*such that

Therefore, the algebraic multiplicity of the eigenvalue $\frac{1}{{\lambda}_{n}^{3}}$ is 1. □

**Lemma 2.3** *Suppose that* (H_{1}) *holds and* $y\in P\mathrm{\setminus}\{\theta \}$ *is a solution of the* (1.1), *then* $y\in \stackrel{\circ}{P}$. *Similarly*, *if* $y\in (-P)\mathrm{\setminus}\{\theta \}$ *is a solution of the* (1.1), *then* $y\in -\stackrel{\circ}{P}$.

*Proof* The proof is obvious. □

**Lemma 2.4** *Suppose that* (H_{1})-(H_{3}) *hold*. *Then the operator* *A* *is Fréchet differentiable at* *θ* *and* ∞, *and* ${A}_{\theta}^{\prime}={\beta}_{0}K$, ${A}_{\mathrm{\infty}}^{\prime}={\beta}_{1}K$.

*Proof*Since (H

_{3}): ${lim}_{u\to 0}\frac{f(t,u)}{u}={\beta}_{0}$ uniformly on $[0,1]$. That is, for any $\epsilon >0$, there exists $\delta >0$ such that

_{1}), it is easy to see that $\mathrm{\forall}t\in [0,1]$, $f(t,0)=0$. Then, for any $u\in E$ with $\parallel u\parallel <\delta $, we have

which means ${A}_{\theta}^{\prime}={\beta}_{0}K$.

_{2}): ${lim}_{|u|\to +\mathrm{\infty}}\frac{f(t,u)}{u}={\beta}_{1}$ uniformly on $[0,1]$. That is, for any $\epsilon >0$, there exists $R>0$ such that

Therefore, ${A}_{\mathrm{\infty}}^{\prime}={\beta}_{1}K$. □

**Remark 5**Suppose (H

_{2}) holds. Similar to Lemma 2.4, we have

**Lemma 2.5** [13]

*Suppose that*

*E*

*is an ordered Banach space with a lattice structure*,

*P*

*is a normal solid cone in*

*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*${A}_{P}^{\prime}$*and*${A}_{-P}^{\prime}$*exist with*$r({A}_{P}^{\prime})>1$*and*$r({A}_{-P}^{\prime})>1$; 1*is not an eigenvalue of*${A}_{P}^{\prime}$*or*${A}_{-P}^{\prime}$*corresponding a positive eigenvector*; - (iii)
$A\theta =\theta $;

*the Fréchet derivative*${A}_{\theta}^{\prime}$*of**A**at**θ**is strongly positive*,*and*$r({A}_{\theta}^{\prime})<1$; - (iv)
*the Fréchet derivative*${A}_{\mathrm{\infty}}^{\prime}$*of**A**at*∞*exists*; 1*is not an eigenvalue of*${A}_{\mathrm{\infty}}^{\prime}$;*the sum**β**of the algebraic multiplicities for all eigenvalues of*${A}_{\mathrm{\infty}}^{\prime}$*lying in the interval*$(1,\mathrm{\infty})$*is an even number*.

*Then* *A* *has at least three nontrivial fixed points containing one sign*-*changing fixed point*.

## 3 Main result

We state the main result of this paper.

**Theorem 3.1** *Suppose that* (H_{1})-(H_{3}) *hold*. *Then the problem* (1.1) *has at least three solutions including a sign*-*changing solution*.

We need only to prove that $A=KF$ satisfies the four conditions of Lemma 2.5.

*Proof*Noticing

- (i)
*A*is strongly increasing on*P*and −*P*. In fact, from (H_{1}) and $K(P\setminus \{\theta \})\subseteq \stackrel{\circ}{P}$, we see that*A*is strongly increasing on*P*. Similarly,*A*is strongly increasing on −*P*. - (ii)
From ${A}_{P}^{\prime}={A}_{-P}^{\prime}={\beta}_{1}K$, ${\lambda}_{2{n}_{1}}^{3}<{\beta}_{1}<{\lambda}_{2{n}_{1}+1}^{3}$ and Lemma 2.2, we find that 1 is not an eigenvalue of ${A}_{P}^{\prime}$ or ${A}_{-P}^{\prime}$ and $r({A}_{P}^{\prime})=r({A}_{-P}^{\prime})=\frac{{\beta}_{1}}{{\lambda}_{1}^{3}}>1$.

- (iii)
From ${A}_{\theta}^{\prime}={\beta}_{0}K$, $0<{\beta}_{0}<{\lambda}_{1}^{3}$, $K(P\setminus \{\theta \})\subseteq \stackrel{\circ}{P}$, Lemma 2.2 and (H

_{1}), we have ${A}_{\theta}^{\prime}$ is strongly positive, $A\theta =\theta $ and $r({A}_{\theta}^{\prime})=\frac{{\beta}_{0}}{{\lambda}_{1}^{3}}<1$. - (iv)
Since ${A}_{\mathrm{\infty}}^{\prime}={\beta}_{1}K$, ${\lambda}_{2{n}_{1}}^{3}<{\beta}_{1}<{\lambda}_{2{n}_{1}+1}^{3}$ and Lemma 2.2, the condition (iv) of Lemma 2.5 is satisfied.

Therefore, from Lemma 2.5, we see that *A* has at least three nontrivial fixed points including one sign-changing fixed point. Then, the problem (1.1) has at least three solutions, including one sign-changing solution. □

**Example 3.1**Consider the following third-order boundary-value problem

By simple calculations, we have ${\lambda}_{1}\approx 3.0167$, ${\lambda}_{2}\approx 6.6506$, ${\lambda}_{3}\approx 10.2782$, ${\beta}_{0}=3$, ${\beta}_{1}=300$. Then it is easy to see that $f(t,u)$ satisfies the conditions (H_{1})-(H_{3}). Therefore, the boundary-value problem (3.1) has at least three solutions, including one sign-changing solution.

## Declarations

### Acknowledgements

The authors are highly grateful for the referees’ careful reading and comments on this paper. The research is supported by Program for Scientific research innovation team in Colleges and universities of Shandong Province, the Doctoral Program Foundation of Education Ministry of China (20133705110003), the Natural Science Foundation of Shandong Province of China (ZR2011AM008, ZR2012AM010, ZR2012AQ024).

## Authors’ Affiliations

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