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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 - \(\Delta^{\nu} y(t)=f(t+\nu - 1,y(t+\nu - 1)), y(\nu - 2)=g(y), y(\nu +b)=0\), where \(f:[\nu-1,\dots,\nu+b-1]_{\mathbb{N}_{\nu-2}}\) is continuous, \(g:\mathcal C([\nu-2,\nu+b]_{\mathbb{N}_{\nu-2}},\mathbb{R})\) is a given functional, and \(1<\nu \leq 2\). 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
39A12 Discrete version of topics in analysis
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