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} $ .

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