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

Our concern is the existence of positive solutions of the discrete Neumann boundary value problem {−Δ2u(t−1)=f(t,u(t)),t∈[1,T]Z,Δu(0)=Δu(T)=0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{aligned} \left \{ \textstyle\begin{array}{@{}l} -\Delta^{2} u(t-1)=f(t, u(t)), \quad t\in[1,T]_{\mathbb{Z}},\\ \Delta u(0)=\Delta u(T)=0, \end{array}\displaystyle \right . \end{aligned}$$ \end{document} where f:[1,T]Z×R+→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$f: [1,T]_{\mathbb{Z}}\times\mathbb{R}^{+}\to\mathbb{R}$\end{document} 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)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$f(t,z)$\end{document} may be unbounded below or nonpositive for all (t,z)∈[1,T]Z×R+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(t,z)\in[1,T]_{\mathbb{Z}}\times\mathbb{R}^{+}$\end{document}.