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On a discrete fractional three-point boundary value problem. (English) Zbl 1253.26010

This author studies the three-point nonlinear discrete fractional boundary value problem \[ -\triangle ^{\nu }y(t) = f(t+\nu -1,y(t+\nu -1)), \]
\[ y(\nu -2) = 0, \]
\[ \alpha y(\nu +K) = y(\nu +b), \] where \(t\in [0,b]_{\mathbb{N}_{0}},\nu \in (1,2],\alpha \in [0,1],K\in [-1,b-1]_{\mathbb{Z}}\) and \(f:[\nu -1,\nu +b-1]_{\mathbb{N}_{\nu -1}}\times \mathbb{R}\rightarrow \mathbb{R}\) is continuous. The Green’s function associated to the above problem is derived and its properties are investigated. By using the Brouwer fixed point theorem and the Krasnoselski fixed point theorem, it is shown that a solution to the above problem exists. The results obtained in this paper generalize some recent results obtained by F. M. Atıcı and P. W. Eloe [“Two-point boundary value problems for finite fractional difference equations”, J. Difference Equ. Appl. 17, No. 4, 445–456 (2011; Zbl 1215.39002)].

MSC:

26A33 Fractional derivatives and integrals
39A05 General theory of difference equations
47H10 Fixed-point theorems

Citations:

Zbl 1215.39002
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References:

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