• 数学 •

### 二阶离散周期边值问题的Ambrosetti-Prodi结果

1. 西北师范大学 数学与统计学院, 兰州　730070
• 收稿日期:2020-07-03 出版日期:2021-11-25 发布日期:2021-11-26
• 通讯作者: 路艳琼 E-mail:393023294@qq.com;luyq8610@126.com;1269469254@qq.com
• 基金资助:
国家自然科学基金青年基金(11901464, 11801453)

### Ambrosetti-Prodi results for second-order discrete periodic boundary value problems

Rui WANG(), Yanqiong LU*(), Xiaomei YANG()

1. College of Mathematics and Statistics, Northwest Normal University, Lanzhou　730070, China
• Received:2020-07-03 Online:2021-11-25 Published:2021-11-26
• Contact: Yanqiong LU E-mail:393023294@qq.com;luyq8610@126.com;1269469254@qq.com

Abstract:

This paper explores the relationship between the number of solutions and the parameter $s$ of second-order discrete periodic boundary value problems of the form　　　　　　　　　　 $\left\{ \begin{array}{ll} \Delta^{2} u(t-1)+f\Delta u(t)+g(t,u(t)) = s, \;t\in[1,T]_{\mathbb{Z}}, \\ u(0) = u(T-1),\;\Delta u(0) = \Delta u(T-1), \end{array} \right.$ where $g: [1,T]_{\mathbb{Z}}\times \mathbb{R}\to\mathbb{R}$ is a continuous function, $f\geqslant0$ is a constant, $T\geqslant2$ is an integer, and $s$ is a real number. By using the upper and lower solution method and the theory of topological degree, we obtain the Ambrosetti-Prodi type alternatives which demonstrate the existence of either zero, one, or two solutions depending on the choice of the parameter $s$ with fixed constant $s_{0}\in \mathbb{R}$ .