Existence and uniqueness of solutions for boundary value problem of fractional order difference equations

In this article, we investigate the existence and uniqueness of solutions for boundary value problem (BVP) of fractional order difference equations (FODE) of the form Δ υ w ( s ) = − f ( s + υ − 1 , w ( s + υ − 1 ) ) , w ( υ − 2 ) = ψ ( w ) , w ( υ + k ) = ϕ ( w ) , where s ∈ [0, k] N 0 , f : [ υ − 2, υ − 1, …, υ + k] N υ −2 × R [0, + ∞] is a continuous function ψ , φ : C ([ υ − 2, υ + k ] N υ −2 ) → R are given functions and 1 < υ ⩽ 2 . Existence and uniqueness of solutions are established by the Brouwer fixed point theorem and contraction mapping principle.