Nonlocal difference equations with sign-changing coefficients - Goodrich, Christopher S.; Lyons, Benjamin (Applied mathematics letters, 2020)
We consider second-order difference equations of the form −A∑j=1Nαj(utj)qΔ2u(n)=λf(n,u(n+1))subject to the Dirichlet boundary conditions u(0)=0=u(b+2). We demonstrate that using a nonstandard cone and associated open set can allow one to deduce the existence of at least one positive solution even in the case where the function A may change sign. Jensen’s inequality plays an important role in our analysis.