Existence and uniqueness of solutions to a fractional difference equation with nonlocal conditions

In this paper, we consider a discrete fractional boundary value problem of the form −Δνy(t)=f(t+ν−1,y(t+ν−1))−Δνy(t)=f(t+ν−1,y(t+ν−1)), y(ν−2)=g(y)y(ν−2)=g(y), y(ν+b)=0y(ν+b)=0, where f:[ν−1,…,ν+b−1]Nν−2×R→Rf:[ν−1,…,ν+b−1]Nν−2×R→R is continuous, g:C([ν−2,ν+b]Nν−2,R)g:C([ν−2,ν+b]Nν−2,R) is a given functional, and 1<ν≤21<ν≤2. We give a representation for the solution to this problem. Finally, we prove the existence and uniqueness of solution to this problem by using a variety of tools from nonlinear functional analysis including the contraction mapping theorem, the Brouwer theorem, and the Krasnosel’skii theorem.