Green's functions for fractional difference equations with Dirichlet boundary conditions. (December 2021) - Cabada, Alberto; Dimitrov, Nikolay D.; Jonnalagadda, Jagan Mohan
Highlights: On the paper is obtained the general solution of a second order discrete fractional equation. Two fractional derivatives are considered on the equation. This is a novelty with respect previous paper in this area. To this end, two functions given as suitable series, are defined. Such functions characterize the vector space of the solutions of the general problem. By considering Dirichlet boundary conditions, the expression of the Green's function is attained. Moreover it is showed a strong positiveness condition of this function. Some direct applications to ensure the existence of solutions of nonlinear boundary problems are deduced. Abstract: This article is devoted to deduce the expression and the main properties of the Green's function related to a general constant coefficients fractional difference equation coupled to Dirichlet conditions. In particular, it is proved that such function has constant sign on their set of definition and, moreover, it satisfies some additional strong sign conditions that are fundamental to define suitable cones, where to ensure the existence of solutions of nonlinear problems.