Double positive solutions of \((n,p)\) boundary value problems for higher order difference equations (Q1352777)

The authors provide existence criteria for double positive solutions of the \((n,p)\) boundary value problem \(\Delta^n y+ F(k,y,\Delta y,\ldots,\Delta^{n-2}y)=G(k,y,\Delta y,\ldots,\Delta^{n-1} y)\) for \(n-1\leq k\leq N\), \(\Delta^i y(0)=0\) for \(0\leq i\leq n-2\), and \(\Delta^p y(N+n-p)=0\), where \(n\geq2\) and \(0\leq p\leq n-1\) is fixed. Upper and lower bounds for the two positive solutions are also established for a particular boundary value problem when \(n=2\). The importance of the obtained results is illustrated by several examples. The technique used here depends on the fixed point theory of operators on a cone as developed by \textit{M. A. Krasnosel'skij} [Positive solution of operator equations 1964; Zbl 0121.10604)] and \textit{D. Guo} and \textit{V. Lakshmikantham} [Nonlinear problems in abstract cones (1988; Zbl 0661.47045)].