## Existence and Multiplicity of Solutions for a 2$n$th-Order Discrete Boundary Value Problems with a Parameter

Wang Zhenguo,

School of Mathematics and Statistics, Huanghuai University, Henan Zhumadian 463000

 基金资助: 国家自然科学基金.  11971126吕梁市科学技术重点研发项目.  Rc2020213

 Fund supported: the NSFC.  11971126the Program for Scientific and Technological Research of Lüliang City.  Rc2020213

Abstract

In this paper, we consider the existence and multiplicity of solutions for a $2n$th-order discrete boundary value problems depending on a parameter $\lambda$. When $\lambda\in\left(\frac{p(T)}{2B}, \frac{1}{2A}\right)$, we obtain a sufficient condition for the existence of solutions of a discrete boundary value problems by means of critical point theory. Finally, one example is given to illustrate our main result.

Keywords： Difference equation ; Boundary value problems ; Critical point theory ; Nontrivial solutions

Wang Zhenguo. Existence and Multiplicity of Solutions for a 2$n$th-Order Discrete Boundary Value Problems with a Parameter. Acta Mathematica Scientia[J], 2022, 42(3): 760-766 doi:

## 1 引言

$\begin{eqnarray} \left\{ \begin{array}{l} \Delta^{n}(p(k)\Delta^{n} u(k-n))+\lambda(-1)^{n+1} f(k, u(k))=0, k\in{\Bbb Z}(1, T), \\ u(0)=u(-1)=\cdots=u(1-n)=0, \Delta^{n}u(T)=\Delta^{n-1}u(T)=\cdots=\Delta u(T)=0, \end{array} \right. \end{eqnarray}$

## 2 相关定义和结论

$\begin{eqnarray} \|u\|=\bigg(\sum^{T}_{k=1}(|u(k)|^{2}\bigg)^{\frac{1}{2}}. \end{eqnarray}$

$\begin{eqnarray} \|u\|_{\infty}=\max\limits_{k\in{\Bbb Z}(1, T)}\{|u(k)|\}. \end{eqnarray}$

$E$是一个自反的实Banach空间, 泛函$I_{\lambda}:E\rightarrow {\Bbb R}$满足下面的结构假设:

(H1) 假设$\lambda$是一个正实参数. 设$I_{\lambda}:=\Phi(u)-\lambda\Psi(u)$, $\forall u\in E$, 其中$\Phi, \Psi\in C^{1}(E, {\Bbb R})$, $\Phi$是强制的, 即$\lim\limits_{\| u\|\rightarrow \infty}\Phi(u)=+\infty$.

$\rm (a)$对任给$r>\inf_{E}\Phi$, 任意的$\lambda\in\left(0, \frac{1}{\varphi(r)}\right)$, 泛函$I_{\lambda}=\Phi(u)-\lambda\Psi(u) $$u\in\Phi^{-1}(-\infty, r) 上有一个全局最小点, 它是 I_{\lambda}$$ E$中局部极小临界点.

$\rm (b)$如果$\gamma<+\infty$, 对任意的$\lambda\in(0, \frac{1}{\gamma})$, 则下列两个结论二者选一:

$\rm (b1) $$I_{\lambda} 存在一个全局最小点, 或者 \rm (b2) 存在 I_{\lambda} 的一个临界点(局部极小点)序列 \{u_{n}\} 使得 \lim\limits_{n\rightarrow \infty}\Phi(u_{n})=+\infty . 下面给出问题(1.1) 的变分框架. 对任意 u\in S , 令 \begin{eqnarray} \Phi(u)=\frac{1}{2}\sum\limits^{T}_{k=1}p(k)(\Delta^{n} u(k-n))^{2}, \Psi(u)=\sum^{T}_{k=1}F(k, u(k)) , I_{\lambda}(u)=\Phi(u)-\lambda\Psi(u), \end{eqnarray} 其中 F(k, \xi)=\int_{0}^{\xi}f(k, s){\rm d}s. 显然, I_{\lambda}\in C^{1}(S, {\Bbb R}) , 直接计算 I_{\lambda}$$ u(k)$偏导数, 得

$$$\frac{\partial I_{\lambda}(u)}{\partial u(k)}=(-1)^{n}\Delta^{n}(p(k)\Delta^{n} u(k-n))-\lambda f(k, u(k)), k\in{\Bbb Z}(1, T).$$$

(H3) $B=\limsup\limits_{\xi\rightarrow +\infty} \frac{F(k, \xi)}{\xi^{2}}$, $\forall k\in{\Bbb Z}(1, T)$.

$$$p_{\ast}=\min\limits _{k\in{\Bbb Z}(1, T)}\{p(k)\}.$$$

$$$|a_{j}|<c_{j},$$$

$$$A<\frac{B}{p(T)},$$$

我们注意到

$\begin{eqnarray} \varphi(r_{j})&\leq&\inf\limits_{\|u\|\leq (\frac{r_{j}}{\alpha})^{1/2}}\frac{\sup\limits_{(\|u\|\leq\frac{r_{j}}{\alpha})^{1/2}} \sum\limits^{T}_{k=1}F(k, u(k)) -\sum\limits^{T}_{k=1} F(k, u(k))}{r_{j}-\Phi(u)}{}\\ &\leq&\inf\limits_{\|u\|\leq(\frac{r_{j}}{\alpha})^{1/2}}\frac{\sum\limits^{T}_{k=1} \max\limits_{|u(k)| \leq c_{j}}F(k, u(k)) -\sum\limits^{T}_{k=1} F(k, u(k))}{\alpha c_{j}^{2}-\Phi(u)}. \end{eqnarray}$

$S$上取一个序列$\{s_{j}\}$满足: 对任意的$k\in{\Bbb Z}(1, T-1)$, 取$(s_{j})(k)=0$, $(s_{j})(T)=d_{j}$. 于是

$T=3$, 对任意的$k\in{\Bbb Z}(1, 3)$, $p(k)=1$, 取充分小的$\varepsilon>0$, 总有$\frac{1}{4+2\varepsilon}< \frac{\alpha}{3\varepsilon}$, 因此, 对任每一个$\lambda\in\left(\frac{1}{4+2\varepsilon}, \frac{\alpha}{3\varepsilon}\right)$, 由推论2.1, 问题(1.1) 存在一个无界无穷解序列.

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