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Erschienen in:
31.03.2016 | Original Research
Existence and uniqueness of solutions to a fractional difference equation with p-Laplacian operator
Erschienen in: Journal of Applied Mathematics and Computing | Ausgabe 1-2/2017
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Abstract
In this paper, we consider a discrete fractional boundary value problem with p-Laplacian operator of the form where \(f:[\alpha +\beta -4,\alpha +\beta +b]_{{\mathbb {N}}_{\alpha +\beta -4}}\times {\mathbb {R}}\rightarrow {\mathbb {R}}\) is a continuous function, and \(p>1\), \(1<\alpha ,\beta \le 2\). We study the existence and uniqueness of solution to this problem by using a variety of tools from nonlinear functional analysis including the contraction mapping theorem and Brouwer fixed point theorem.
$$\begin{aligned} {\left\{ \begin{array}{ll}\Delta ^{\beta }[\phi _{p}(\Delta ^{\alpha }y)](t)+f(\alpha +\beta +t-1,y(\alpha +\beta +t-1))=0, t\in [0,b]_{{\mathbb {N}}_{0}},\\ \Delta ^{\alpha }y(\beta -2)=\Delta ^{\alpha }y(\beta +b)=0,\\ y(\alpha +\beta -4)=y(\alpha +\beta +b)=0,\end{array}\right. } \end{aligned}$$