Existence to singular discrete boundary value problems with sign changing nonlinearities using approximation methods

Abstract

In this paper we use approximation methods to examine the singular discrete boundary value problem \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\left\{ \begin{gathered} - \Delta ^2 u(k - 1) = g(k,u(k)) + h(k,u(k)),k \in [1,T] \hfill \\ u(0) = 0 = u(T + 1) \hfill \\ \end{gathered} \right.$$ \end{document} where our nonlinearity may be singular in its dependent variable and is allowed to change sign.