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.


39A10 Additive difference equations
26A33 Fractional derivatives and integrals
39A12 Discrete version of topics in analysis
Full Text: DOI


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