EXISTENCE AND UNIQUENESS OF SOLUTIONS FORBOUNDARY VALUE PROBLEMS TO SYSTEM OF FRACTIONALDIffERENCE EQUATIONS

Abstract

In this work, we focus on the existence and uniqueness of solutions for the following system of fractional differential equations: 
​Δ^αx(k)=f(k+α−1,x(k+α−1),y(k+β−1))
Δ^βy(k)=g(k+β−1,x(k+α−1),y(k+β−1))
α_0​x(α−1)−β₀​Δx(α−2)=0
γ₀​x(α+b)+δ_0​Δx(α−2)=0
αˉ₀​y(β−1)−βˉ​₀​Δy(β−2)=0
γˉ​₀​y(β+b)+δˉ_0​Δy(β−2)=0​ 
Where k∈N(0,b),1<α,β≤2,f,g:N(0,b)×R×R→R are continuous functions and α₀​γ₀​+α₀​δ₀​+β₀​γ₀​≠0,  αˉ₀​​ γˉ​₀​_​+αˉ₀​​ δˉ₀​+βˉ​₀γˉ​₀ ​≠ 0.
α_ο, β_ο, γ_ο, αˉ_ο, βˉ_ο, ˉ​γ_ο, δˉ_0​∈R⁺. We derive a representation for the solution and establish the uniqueness of the solution by employing the Perov's fixed-point