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Signchanging solution for a thirdorder boundaryvalue problem in ordered Banach space with lattice structure
Boundary Value Problemsvolume 2014, Article number: 132 (2014)
Abstract
In this paper, the signchanging solution of a thirdorder twopoint boundaryvalue 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 signchanging solution.
1 Introduction
In this paper, we consider the following nonlinear thirdorder twopoint boundaryvalue problem
where $f\in C([0,1]\times R,R)$.
The study on the existence of the signchanging solutions for the boundaryvalue problem is very useful and interesting both in theory and in application. Recently, there has been much attention focused on the problem, especially to the twopoint or multipoint boundaryvalue problem. For the secondorder twopoint or multipoint boundaryvalue problem, many beautiful results have been given on the existence and multiplicity of the signchanging solutions (see [1–5] and the references therein). For example, Xu and Sun [1] obtained an existence result of the signchanging solutions for the secondorder threepoint boundaryvalue problem
where $0<\alpha <1$, $0<\eta <1$, $f\in C(R,R)$. Xu [2] considered the signchanging solutions for the secondorder multipoint boundaryvalue problem
where ${\alpha}_{i}>0$, $i=1,2,\dots ,m2$, $0<{\eta}_{1}<\cdots <{\eta}_{m2}<1$, $f\in C(R,R)$. In [4], Zhang and Sun obtained the existence and multiplicity of the signchanging solutions for the integral boundaryvalue problem
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 boundaryvalue problem (1.2), Li and Liu [5] also obtained the existence and multiplicity of the signchanging solutions in ordered Banach space with the lattice structure.
For the thirdorder boundaryvalue 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 signchanging 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 signchanging 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 signchanging solutions of the problem (1.1).
The main contribution of this paper are as follows: (a) for the signchanging 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 signchanging 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, $\{u\in E:u(t)>0,t\in (0,1]\}\subset \stackrel{\circ}{P}=\{x\in Px\text{is an interior point of}P\}$.
Let the operators K, F, A be defined by
and $A=KF$, respectively, where
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]
Let 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]
Let $D\subseteq E$ and $A:D\u27f6E$ be a nonlinear operator. A is said to be quasiadditive 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 quasiadditive 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 $t\in [0,1]$, $u\in R\mathrm{\setminus}\{0\}$.
(H_{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
where $0<{\lambda}_{1}<{\lambda}_{2}<\cdots <{\lambda}_{n}<\cdots $ are the positive solutions of the equation
(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
over $[0,t]$ for three times, we have
Then
Combining them with boundary condition $u(0)={u}^{\prime}(0)={u}^{\prime}(1)=0$, we conclude that
Therefore,
On the other hand, since
therefore,
and
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
The auxiliary equation of the differential equation (2.4) has roots −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
Then
Applying the condition $y(0)={y}^{\prime}(0)=0$, we obtain ${C}_{2}={C}_{1}$, ${C}_{3}=\sqrt{3}{C}_{1}$, where ${C}_{1}\ne 0$.
Applying the second condition ${y}^{\prime}(1)=0$, that is,
Considering (2.3), we see that μ is one of ${\lambda}_{1},{\lambda}_{2},\dots ,{\lambda}_{n},\dots $ , that is,
are eigenvalues of the linear operator K and the eigenfunction corresponding to the eigenvalue $\frac{1}{{\lambda}_{n}^{3}}$ is
where C is a nonzero constant.
Next we prove that the algebraic multiplicity of the eigenvalue $\frac{1}{{\lambda}_{n}^{3}}$ is 1. From (2.6), any two eigenfunctions corresponding to the same eigenvalue $\frac{1}{{\lambda}_{n}^{3}}$ are merely nonzero constant multiples of each other, that is,
Now we show that
Obviously, we only need to show that
In fact, for any $y\in ker{(I{\lambda}_{n}^{3}K)}^{2}$, $(I{\lambda}_{n}^{3}K)y$ is an eigenfunction of linear operator 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
By direct computation, we have
It is easy to see that the solution for the corresponding homogeneous equation of (2.7) is of the form
Then, by an ordinary differential equation method, we see that the general solution of (2.7) is of the form
where
is the special solution of the equation
and
is the special solution of the equation
Then
Applying the condition $y(0)={y}^{\prime}(0)=0$, we obtain ${C}_{2}={C}_{1}$, ${C}_{3}=\sqrt{3}{C}_{1}$. From the condition ${y}^{\prime}(1)=0$, we obtain
From (2.5), we have ${({y}_{0})}^{\prime}(1)=0$. Thus it follows from (2.8) that
which implies that
That is
which is a contradiction of
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
From (H_{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
Then
Thus,
which means ${A}_{\theta}^{\prime}={\beta}_{0}K$.
Since (H_{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
Let ${M}_{R}={max}_{0\le u\le R}f(t,u)$. Then
Thus,
Then
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 quasiadditive 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 signchanging 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 signchanging 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 signchanging fixed point. Then, the problem (1.1) has at least three solutions, including one signchanging solution. □
Example 3.1 Consider the following thirdorder boundaryvalue problem
where
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 boundaryvalue problem (3.1) has at least three solutions, including one signchanging solution.
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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).
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Keywords
 thirdorder boundaryvalue problem
 signchanging solution
 Green’s function
 fixed point
 lattice