Study on the Existence of Sign-Changing Solutions of Case Theory Based a Class of Differential Equations Boundary-Value Problems

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1. Introduction

The existence of nonlinear three-point boundary-value problems has been studied [1] - [6] , and the existence of sign-changing solutions is obtained. In the past, most studies were focused on the cone fixed point index theory [7] [8] [9] , just a few took use of case theory to study the topological degree of non-cone mapping and the calculation of fixed point index, and the case theory was combined with the topological degree theory to study the sign-changing solutions. Recent study Ref. [10] [11] have given the method of calculating the topological degree under the case structure, and taken use of the fixed point theorem of non-cone mapping to study the existence of nontrivial solutions for the nonlinear Sturm-Liouville problems. Relevant studies as [12] [13] [14] .

Inspired by the Ref. [8] - [13] and by using the new fixed point theorem under the case structure, this paper studies three-point boundary-value problems for A class of nonlinear second-order equations

$\{\begin{array}{l}{u}^{\u2033}\left(t\right)+f\left(u\left(t\right)\right)=0,\text{\hspace{0.17em}}0\le t\le 1;\\ {u}^{\prime}\left(0\right)=0,\text{\hspace{0.17em}}u\left(1\right)=\alpha u\left(\eta \right),\end{array}$ (1)

Existence of the sign-changing solution, constant $0<\alpha <1,0<\eta <1$ , $f\in C\left(R,R\right)$ .

Boundary-value problem (1) is equivalent to Hammerstein nonlinear integral equation hereunder

$u\left(t\right)={\displaystyle {\int}_{0}^{1}G\left(t,s\right)f\left(u\left(s\right)\right)\text{d}s},\text{\hspace{0.17em}}0\le t\le 1$ (2)

Of which $G\left(t,s\right)$ is the Green function hereunder

$G\left(t,s\right)=\frac{1}{1-\alpha}\{\begin{array}{l}\left(1-s\right)-\alpha \left(\eta -s\right),0\le s\le \eta ,0\le t\le s;\\ \left(1-s\right),\eta \le s\le 1,0\le t\le s;\\ \left(1-\alpha \eta \right)-t\left(1-\alpha \right),0\le s\le \eta ,s\le t\le 1;\\ \left(1-\alpha y\right)-t\left(1-\alpha \right),\eta \le s\le 1,s\le t\le 1.\end{array}$

Defining linear operator K as follow

$\left(Ku\right)\left(t\right)={\displaystyle {\int}_{0}^{1}G\left(t,s\right)u\left(s\right)\text{d}s},\text{\hspace{0.17em}}u\in C\left[0,1\right].$ (3)

Let $Fu\left(t\right)=f\left(u\left(t\right)\right)$ , $t\in \left[0,1\right]$ , obviously composition operator $A=KF$ , i.e.

$\left(Au\right)\left(t\right)={\displaystyle {\int}_{0}^{1}G\left(t,s\right)f\left(u\left(s\right)\right)\text{d}s},\text{\hspace{0.17em}}0\le t\le 1$ (4)

It’s easy to get: $u\in {C}^{2}\left[0,1\right]$ is the solution of boundary-value problem (1), and $u\in C\left[0,1\right]$ is the solution of operator equation $u=Au$ .

We note that, in Ref. [9] [10] , an abstract result on the existence of sign- changing solutions can be directly applied to problem (1). After the necessary preparation, when the non-linear term $f$ is under certain assumptions, we get the existence of sign-changing solution of such boundary-value problems. Compared with the Ref. [8] , we can see that we generalize and improve the nonlinear term $f$ , and remove the conditions of strictly increasing function, and the method is different from Ref. [8] .

For convenience, we give the following conditions.

(H_{1})
$f\left(u\right):R\to R$ continues,
$f\left(u\right)u>0$ ,
$\forall u\in R,u\ne 0$ , and
$f\left(0\right)=0$ .

(H_{2})
$\underset{u\to 0}{\mathrm{lim}}\frac{f\left(u\right)}{u}=\beta $ , and
${n}_{0}\in N$ , make
${\lambda}_{2{n}_{0}}<\beta <{\lambda}_{2{n}_{0}+1}$ , of which
$0<{\lambda}_{1}<{\lambda}_{2}<\cdots <{\lambda}_{n}<{\lambda}_{n+1}<\cdots $ is the positive sequence of
$\mathrm{cos}\sqrt{x}=\alpha \mathrm{cos}\eta \sqrt{x}$ .

(H_{3}) exists
$\epsilon >0$ , make
$\underset{\left|u\right|\to +\infty}{\mathrm{lim}}\mathrm{sup}\frac{f\left(u\right)}{u}\le {\lambda}_{1}-\epsilon $ .

2. Knowledge

Provided P is the cone of E in Banach space, the semi order in E is exported by cone P. If the constant $N>0$ , and $\theta \le x\le y\Rightarrow \Vert x\Vert \le N\Vert y\Vert $ , then P is a normal cone; if P contains internal point, i.e. $\mathrm{int}P\ne \varnothing $ , then P is a solid cone.

E becomes a case when semi order £, i.e. any $x,y\in E$ , $\mathrm{sup}\left\{x,y\right\}$ and $\mathrm{inf}\left\{x,y\right\}$ is existed, for $x\in E$ , ${x}^{+}=\mathrm{sup}\left\{x,\theta \right\}$ , ${x}^{-}=\mathrm{sup}\left\{-x,\theta \right\}$ , we call positive and negative of x respectively, call $\left|x\right|={x}^{+}+{x}^{-}$ as the modulus of x. Obviously, ${x}^{+}\in P$ , ${x}^{-}\in \left(-P\right)$ , $\left|x\right|\in P$ , $x={x}^{+}-{x}^{-}$ .

For convenience, we use the following signs: ${x}_{+}={x}^{+}$ , ${x}_{-}=-{x}^{-}$ . Such that $x={x}_{+}+{x}_{-}$ , $\left|x\right|={x}_{+}-{x}_{-}$ .

Provided Banach space $E=C\left[0,1\right]$ , and E’s norm as $\Vert \text{\hspace{0.05em}}\cdot \text{\hspace{0.05em}}\Vert $ , i.e.

$\Vert u\Vert =\underset{0\le t\le 1}{\mathrm{max}}\left|u\left(t\right)\right|$ . Let $P=\left\{u\in E|u\left(t\right)\ge 0,t\in \left[0,1\right]\right\}$ , then P is the normal cone of

E, and E becomes a case under semi order £.

Now we give the definitions and theorems

Def 1 [10] provided $D\subset E,A:D\to E$ is an operator (generally a nonlinear). If $Ax=A{x}_{+}+A{x}_{-},\forall x\in E$ , then A is an additive operator under case structure; if ${v}^{\ast}\in E$ , and $Ax=A{x}_{+}+A{x}_{-}+{v}^{\ast},\forall x\in E$ , then A is a quasi additive operator.

Def 2 provided x is a fixed point of A, if $x\in \left(P\backslash \left\{\theta \right\}\right)$ , then x is a positive fixed point; if $x\in \left(\left(-P\right)\backslash \left\{\theta \right\}\right)$ , then x is a negative fixed point; if $x\notin \left(P\cup \left(-P\right)\right)$ , then x is a sign-changing fixed point.

Lemma 1 [6] $G\left(t,s\right)$ is a nonnegative continuous function of $\left[0,1\right]\times \left[0,1\right]$ ,

and when $t,s\in \left[0,1\right]$ , $G\left(t,s\right)\ge \gamma G\left(0,s\right)$ , of which $\gamma =\frac{\alpha \left(1-\eta \right)}{1-\alpha \eta}$ .

Lemma 2 $K:P\to P$ is completely continuous operator, and $A:E\to E$ is completely continuous operator.

Lemma 3 A is a quasi additive operator under case structure.

Proof: Similar to the proofs in Lemma 4.3.1 in Ref. [10] , get Lemma 3 works.

Lemma 4 [6] the eigenvalues of the linear operator K are

$\frac{1}{{\lambda}_{1}},\frac{1}{{\lambda}_{2}},\cdots ,\frac{1}{{\lambda}_{n}},\frac{1}{{\lambda}_{n+1}},\cdots $ . And the sum of algebraic multiplicity of all eigenvalues is

1, of which
${\lambda}_{n}$ is defined by (H_{2}).

The lemmas hereunder are the main study bases.

Lemma 5 [10] provided E is Banach space, P is the normal cone in E, $A:E\to E$ is completely continuous operator, and quasi additive operator under case structure. Provided that

1) There exists positive bounded linear operator ${B}_{1}$ , and ${B}_{1}$ ’s $r\left({B}_{1}\right)<1$ , and ${u}^{\ast}\in P,{u}_{1}\in P$ , get

$-{u}^{\ast}\le Ax\le {B}_{1}x+{u}_{1},\forall x\in P;$

2) There exists positive bounded linear operator ${B}_{2}$ , ${B}_{2}$ ’s $r\left({B}_{2}\right)<1$ , and ${u}_{2}\in P$ , get

$Ax\ge {B}_{2}x-{u}_{2},\forall x\in \left(-P\right);$

3) $A\theta =\theta $ , there exists Frechet derivative ${{A}^{\prime}}_{\theta}$ of A at $\theta $ , 1 is not the eigenvalue of ${{A}^{\prime}}_{\theta}$ , and the sum $\mu $ of algebraic multiplicity of ${{A}^{\prime}}_{\theta}$ ’s all eigenvalues in the range $\left(1,\infty \right)$ is a nonzero even number,

$A\left(P\backslash \left\{\theta \right\}\right)\subset \stackrel{\xb0}{P},\text{\hspace{0.17em}}\text{\hspace{0.17em}}A\left(\left(-P\right)\backslash \left\{\theta \right\}\right)\subset -\stackrel{\xb0}{P}$

Then A exists three nonzero fixed points at least: one positive fixed point, one negative fixed point and a sign-changing fixed point.

3. Results

Theorem provided (H_{1}) (H_{2}) (H_{3}) works, boundary-value problem (1) exists a sign-changing solution at least, and also a positive solution and a negative solution.

Proof provided linear operator $B=\left({\lambda}_{1}-\frac{\epsilon}{2}\right)K$ , Lemma 2 knows $B:C\left[0,1\right]\to C\left[0,1\right]$ is a positive bounded linear operator. Lemma 4 gets K’s $r\left(K\right)=\frac{1}{{\lambda}_{1}}$ , so $r\left(B\right)=\left({\lambda}_{1}-\frac{\epsilon}{2}\right)r\left(K\right)=1-\frac{\epsilon}{2{\lambda}_{1}}<1$ .

(H_{3}) knows
$m>0$ and gets

$f\left(u\right)\le \left({\lambda}_{1}-\frac{\epsilon}{2}\right)u+m,\text{\hspace{0.17em}}\forall t\in \left[0,1\right],\text{\hspace{0.17em}}u\ge 0$ (5)

$f\left(u\right)\ge \left({\lambda}_{1}-\frac{\epsilon}{2}\right)u-m,\text{\hspace{0.17em}}\forall t\in \left[0,1\right],\text{\hspace{0.17em}}u\le 0$ (6)

Let ${u}_{0}\left(t\right)=m{\displaystyle {\int}_{0}^{1}G\left(t,s\right)\text{d}s}$ , obviously, ${u}_{0}\in P$ . Such that, for any $u\left(t\right)\in P$ ,

there

$\begin{array}{c}\left(Au\right)\left(t\right)={\displaystyle {\int}_{0}^{1}G\left(t,s\right)f\left(u\left(s\right)\right)\text{d}s}\\ \le {\displaystyle {\int}_{0}^{1}G\left(t,s\right)\left(\left({\lambda}_{1}-\frac{\epsilon}{2}\right)u+m\right)\text{d}s}\\ \le \left({\lambda}_{1}-\frac{\epsilon}{2}\right){\displaystyle {\int}_{0}^{1}G\left(t,s\right)u\left(s\right)\text{d}s}+m{\displaystyle {\int}_{0}^{1}G\left(t,s\right)\text{d}s}\\ =\left({\lambda}_{1}-\frac{\epsilon}{2}\right)Ku\left(t\right)+{u}_{0}\left(t\right)\\ =Bu\left(t\right)+{u}_{0}(\; t\; )\end{array}$

And for any
${u}^{\ast}\in P$ , from (H_{1}), obviously gets
$\left(Au\right)\left(t\right)\ge -{u}^{\ast}\left(t\right)$ .

For any $u\left(t\right)\in -P$ , there

$\begin{array}{c}\left(Au\right)\left(t\right)={\displaystyle {\int}_{0}^{1}G\left(t,s\right)f\left(u\left(s\right)\right)\text{d}s}\\ \ge {\displaystyle {\int}_{0}^{1}G\left(t,s\right)\left(\left({\lambda}_{1}-\frac{\epsilon}{2}\right)u-m\right)\text{d}s}\\ \ge \left({\lambda}_{1}-\frac{\epsilon}{2}\right){\displaystyle {\int}_{0}^{1}G\left(t,s\right)u\left(s\right)\text{d}s}-m{\displaystyle {\int}_{0}^{1}G\left(t,s\right)\text{d}s}\\ =\left({\lambda}_{1}-\frac{\epsilon}{2}\right)Ku\left(t\right)-{u}_{0}\left(t\right)\\ =Bu\left(t\right)-{u}_{0}(\; t\; )\end{array}$

Consequently (1) (2) in lemma 5 works.

We note that
$f\left(0\right)=0$ can get
$A\theta =\theta $ , from (H_{2}), we know
$\forall \epsilon >0$ , and
$\exists r>0$ gets

$\left|f\left(u\right)-\beta u\right|\le \epsilon u,\text{\hspace{0.17em}}\left|u\right|\le r$

Then

$\left|\left(Fu\right)\left(t\right)-\lambda u\left(t\right)\right|=\left|f\left(u\left(t\right)\right)-\beta u\left(t\right)\right|\le \epsilon \Vert u\Vert ,\text{\hspace{0.17em}}\forall \Vert u\Vert \le r$

$\Vert Au-A\theta -\beta Ku\Vert =\Vert K\left(Fu\right)-\beta Ku\Vert \le \epsilon \Vert K\Vert \Vert u\Vert ,\text{\hspace{0.17em}}\forall \Vert u\Vert \le r$

Such that

$\underset{\Vert u\Vert \to 0}{\mathrm{lim}}\frac{\Vert Au-A\theta -\beta Ku\Vert}{\Vert u\Vert}=0$

i.e. ${{A}^{\prime}}_{\theta}=\beta K$ , from lemma 4 we get linear operator K’s eigenvalue is $\frac{1}{{\lambda}_{n}}$ , then ${{A}^{\prime}}_{\theta}$ ’s eigenvalue is $\frac{\beta}{{\lambda}_{n}}$ . Because ${\lambda}_{2{n}_{0}}<\beta <{\lambda}_{2{n}_{0}+1}$ , let $\mu $ be the sum of

algebraic multiplicity of ${{A}^{\prime}}_{\theta}$ ’s all eigenvalues in the range $\left(1,\infty \right)$ , then $\mu =2{n}_{0}$ is an even number.

From (H_{1})
$f\left(u\right)u>0$ ,
$u\in R\backslash \left\{0\right\}$ , there

$f\left(u\left(t\right)\right)>0,\text{\hspace{0.17em}}\forall t\in \left[0,1\right],\text{\hspace{0.17em}}u\left(t\right)>0,$

$f\left(u\left(t\right)\right)<0,\text{\hspace{0.17em}}\forall t\in \left[0,1\right],\text{\hspace{0.17em}}u\left(t\right)<0.$

Easy to get

$F\left(P\backslash \left\{\theta \right\}\right)\subset P\backslash \left\{\theta \right\},\text{\hspace{0.17em}}F\left(\left(-P\right)\backslash \left\{\theta \right\}\right)\subset \left(-P\right)\backslash \left\{\theta \right\},$

Lemma (1) for any $u\left(t\right)\in P$ , $\left(Ku\right)\left(t\right)={\displaystyle {\int}_{0}^{1}G\left(t,s\right)u\left(s\right)\text{d}s}\ge \gamma {\displaystyle {\int}_{0}^{1}G\left(0,s\right)u\left(s\right)\text{d}s}$ ,

consequently $K\left(P\backslash \left\{\theta \right\}\right)\subset \stackrel{\xb0}{P}$ . Such that

$A\left(P\backslash \left\{\theta \right\}\right)\subset \stackrel{\xb0}{P},\text{\hspace{0.17em}}A\left(\left(-P\right)\backslash \left\{\theta \right\}\right)\subset -\stackrel{\xb0}{P},$

Such that (3) in lemma 5 works. According to lemma 5, A exists three nonzero fixed points at least: one positive fixed point, one negative fixed point and one sign-changing fixed point. Which states that boundary-value problem (1) has three nonzero solutions at least: one positive solution, one negative solution and one sign-changing solution.

4. Conclusion

Provided that all conditions of the theorem are satisfied, and $f\left(u\right)$ is an odd function, then boundary-value problem (1) has four nonzero solutions at least: one positive solution, one negative solution and two sign-changing solutions.

Note

By using case theory to study the topological degree of non-cone mapping and the calculation of fixed point index, it’s an attempt to combine case theory and topological degree theory, the author thinks it’s an up-and-coming topic and expects to have further progress on that.

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