Positive solutions of discrete Neumann boundary value problems with sign-changing nonlinearities

Abstract

Our concern is the existence of positive solutions of the discrete Neumann boundary value problem $$\begin{aligned} \left \{ \textstyle\begin{array}{@{}l} -\Delta^{2} u(t-1)=f(t, u(t)), \quad t\in[1,T]_{\mathbb{Z}}, \Delta u(0)=\Delta u(T)=0, \end{array}\displaystyle \right . \end{aligned}$$ where $f: [1,T]_{\mathbb{Z}}\times\mathbb{R}^{+}\to\mathbb{R}$ is a sign-changing function. By using the Guo-Krasnosel’skiĭ fixed point theorem, the existence and multiplicity of positive solutions are established. The nonlinear term $f(t,z)$ may be unbounded below or nonpositive for all $(t,z)\in[1,T]_{\mathbb{Z}}\times\mathbb{R}^{+}$ .