Abstract
In this article, we present a two-point boundary value problem with separated boundary conditions for a finite nabla fractional difference equation. First, we construct an associated Green’s function as a series of functions with the help of spectral theory, and obtain some of its properties. Under suitable conditions on the nonlinear part of the nabla fractional difference equation, we deduce two existence results of the considered nonlinear problem by means of two Leray–Schauder fixed point theorems. We provide a couple of examples to illustrate the applicability of the established results.
Highlights
We consider the following nabla fractional difference equation associated with separated boundary conditions: Guirao
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+ ( β − α)γHν−1 (b, ρ(s)) + ( β − α)δHν−2 (b, ρ(s)), v2 (t, s) = v1 (t, s) − Hν−1 (t, ρ(s)), ξ = ( β − α)γ + αγHν−1 (b, a) + αδHν−2 (b, a) 6= 0. This result was obtained by expressing the general solution of the nabla fractional difference equation in (3), using the method of variation of constants
Summary
Denote the set of all real numbers and positive real numbers by R and R+ , respectively. + ( β − α)γHν−1 (b, ρ(s)) + ( β − α)δHν−2 (b, ρ(s)) , v2 (t, s) = v1 (t, s) − Hν−1 (t, ρ(s)), ξ = ( β − α)γ + αγHν−1 (b, a) + αδHν−2 (b, a) 6= 0 This result was obtained by expressing the general solution of the nabla fractional difference equation in (3), using the method of variation of constants. For a non-constant function g the expression of the general solution does not exist and, as a consequence, the method used in [8] is not applicable for the following boundary value problem: Symmetry 2021, 13, 1101 Due to this reason, Graef et al [19] and Cabada et al [20] followed a different approach. We give some examples to demonstrate the applicability of these results
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