Stability and asymptotic properties of a linear fractional difference equation

Abstract

This paper discusses qualitative properties of the two-term linear fractional difference equation 0∇αy(n) = λy(n), where α,λ ε ℝ, 0 < α < 1, λ ≠ 1 and 0∇α is the αth order Riemann-Liouville difference operator. For this purpose, we show that this fractional equation is the Volterra equation of convolution type. This enables us to analyse its qualitative properties by use of tools standardly employed in the qualitative investigation of Volterra difference equations. As the main result, we derive a sharp condition for the asymptotic stability of the studied equation and, moreover, give a precise asymptotic description of its solutions. © 2012 Čermák et al.