Existence Results for Fractional Difference Equations with Three-Point Fractional Sum Boundary Conditions

We consider a discrete fractional boundary value problem of the form Δαu(t)=f(t+α-1,u(t+α-1)),  t∈[0,T]ℕ0:={0,1,…,T},  u(α-2)=0,  u(α+T)=Δ-βu(η+β), where 1<α≤2, β>0, η∈[α-2,α+T-1]ℕα-2:={α-2,α-1,…,α+T-1}, and f:[α-1,α,…,α+T-1]ℕα-1×ℝ→ℝ is a continuous function. The existence of at least one solution is proved by using Krasnoselskii's fixed point theorem and Leray-Schauder's nonlinear alternative. Some illustrative examples are also presented