Title: Some new existence results for fractional difference equations

Authors: Christopher S. Goodrich

Addresses: Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE 68588 USA

Abstract: In this paper, we introduce several existence theorems for a discrete fractional boundary value problem with Dirichlet boundary conditions in the case where the order ν of the fractional difference satisfies 1 < ν ≤ 2. We use cone theoretic techniques to deduce the existence of one or more positive solutions. We then deduce uniqueness theorems for the same problem by assuming a Lipschitz condition. We show that many of the classical existence and uniqueness theorems for second-order discrete boundary value problems extend to the fractional-order case.

Keywords: discrete fractional calculus; contraction mapping theorem; Green|s function; Dirichlet boundary value problem; fixed point theorems; Banach spaces; fractional difference equations.

DOI: 10.1504/IJDSDE.2011.038499

International Journal of Dynamical Systems and Differential Equations, 2011 Vol.3 No.1/2, pp.145 - 162

Published online: 09 Feb 2011 *

Full-text access for editors Access for subscribers Purchase this article Comment on this article