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Existence and uniqueness of solutions to a fractional difference equation with nonlocal conditions. (English) Zbl 1211.39002
Summary: We consider a discrete fractional boundary value problem of the form - ${{\Delta }}^{\nu }y\left(t\right)=f\left(t+\nu -1,y\left(t+\nu -1\right)\right),y\left(\nu -2\right)=g\left(y\right),y\left(\nu +b\right)=0$, where $f:{\left[\nu -1,\cdots ,\nu +b-1\right]}_{{ℕ}_{\nu -2}}$ is continuous, $g:𝒞\left({\left[\nu -2,\nu +b\right]}_{{ℕ}_{\nu -2}},ℝ\right)$ is a given functional, and $1<\nu \le 2$. 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.
##### MSC:
 39A10 Additive difference equations 26A33 Fractional derivatives and integrals (real functions) 39A12 Discrete version of topics in analysis
##### References:
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