An application of topological transversality to non-positive higher order difference equations (Q1294289)

The authors provide sufficient conditions on the non-positive function \(f(i,u_1,\ldots,u_{n-1})\) so that the boundary value problem given by \[ \Delta^n u(i)+\mu f(i,u(i),\Delta u(i),\ldots,\Delta^{n-2}u(i))=0, \] with boundary conditions \[ \Delta^m u(0)=0,\;0\leq m\leq n-3, \quad \alpha\Delta^{n-2}u(0)-\beta\Delta^{n-1} u(0)=0, \] \[ \gamma\Delta^{n-2} u(T+1)+\delta\Delta^{n-1} u(T+1)=0, \] has at least one positive solution. The topological transversality approach used here relies on a priori bounds on the solutions.