Nonlocal difference equations with sign-changing coefficients. (August 2020) - Goodrich, Christopher S.; Lyons, Benjamin
Abstract: We consider second-order difference equations of the form − A ∑ j = 1 N α j ( u t j ) q Δ 2 u ( n ) = λ f ( 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.