The Asymptotic Behavior of Solutions of Discrete Nonlinear Fractional Equations

Abstract

In this study, we consider a class of nonlinear fractional difference equations following form $$\Delta \left( {a\left( t \right)g\left( {{\Delta ^\alpha }x\left( t \right)} \right)} \right) + F\left( {t,\sum\limits_{s = {t_0}}^{t - 1 + \alpha } {{{\left( {t - s - 1} \right)}^{\left( { - \alpha } \right)}}x\left( s \right)} } \right) = 0,$$ where t ∈ Nt0+1−α and Δα denotes the Riemann-Liouville fractional difference operator of order α. Using the generalized Riccati function, we obtain some oscillation criteria. Finally, we give some illustrative examples. MSC 2010: Primary 26A33; Secondary 34A08, 39A21, 39A10