Existence of positive solutions of nonlinear discrete Sturm-Liouville problems
Consider the nonlinear boundary-value problem \[ -\Delta[p(t- 1)\Delta y(t-1)]+ q(t) y(t)= f(t,y(t)),\quad t\in [1,b+1], \] \[ \alpha y(0)- \beta p(0)\Delta y(0)= 0,\quad \gamma y(b+ 1)+\delta p(b+1)\Delta y(b+ 1)= 0, \] where \(b\geq 2\) is a fixed integer in the discrete interval \([1,b+1]\). The author gives the Green function \(G(s,t)\) for the problem under consideration. Let \(A\) be the operator defined by \[ Ay(t):= \sum^{b+1}_{s=1} G(t,s) f(s,y(s)). \] It is shown that under some assumptions the operator \(A\) is differentiable along a certain cone in the Banach space of all nonnegative functions \(y: [0,b+2]\to \mathbb{R}\) satisfying the given boundary conditions. On the basis of this result, the author proves that the discrete Sturm-Liouville boundary-value problem as above but with boundary conditions \[ y(0)= y(b+ 2)= 0, \] has at least one positive solution.
nonlinear discrete Sturm-Liouville problems
scientific article; zbMATH DE number 1545779