On the existence of positive solutions of higher order difference equations (Q1130202)

The authors consider the \(n\)-th order difference equation \[ \Delta^ny+ Q(k,y,\Delta y,\dots,\Delta^{n-2}y)= P(k,y,\Delta y,\dots, \Delta^{n-1}y), \quad k\in[0,N] \] satisfying the boundary conditions \[ \Delta^i y(0)=0, \qquad 0\leq i\leq n-3; \] \[ \alpha\Delta^{n-2} y(0)- \beta\Delta^{n-1} y(0)=0; \qquad\gamma\Delta^{n-2} y(N+1)+ \delta\Delta^{n-1} y(N+1)=0; \] where \(n\geq 2\), \(N(\geq n-1)\) is a fixed positive integer, \(\alpha\), \(\beta\), \(\gamma\) and \(\delta\) are constants satisfying \[ \rho= \alpha\gamma(N+1)+ \alpha\delta+ \beta\gamma> 0; \qquad \alpha>0,\quad \gamma>0,\quad \beta\geq 0,\quad \delta\geq \gamma. \] They state a fixed point theorem due to Krasnoselskij and present some properties of a certain Green function used for providing an appropriate Banach space and a cone so that the fixed point theorem may be applied to yield a positive solution of the difference equation.