Exact multiplicity of solutions for discrete second order Neumann boundary value problems

Our concern is the second order difference equation Δ2u(t−1)+g(u(t))=h(t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\Delta^{2} u(t-1)+g(u(t))=h(t)$\end{document} subject to the Neumann boundary conditions 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}$\Delta u(0)=\Delta u(T)=0$\end{document}. Under convex/concave conditions imposed on g, some results on the exact numbers of solutions and positive solutions are established based on the discussions to the maximum and minimum numbers of (positive) solutions.