Journal of East China Normal University(Natural Science) ›› 2021, Vol. 2021 ›› Issue (6): 47-57.doi: 10.3969/j.issn.1000-5641.2021.06.006

• Mathematics • Previous Articles     Next Articles

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;;


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} $ .

Key words: Ambrosetti-Prodi problem, upper and lower solution method, topological degree theory

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