Oscillation Tests for Fractional Difference Equations

In this paper, we study the oscillatory behavior of solutions of the fractional difference equation of the form Δ ( r ( t ) g ( Δ α x ( t ) ) ) + p ( t ) f ( ∑ s = t 0 t − 1 + α ( t − s − 1 ) ( − α ) x ( s ) ) = 0 , t ∈ t 0 + 1 − α , $$\Delta \left( {r\left( t \right)g\left( {{\Delta ^\alpha }x(t)} \right)} \right) + p(t)f\left( {\sum\limits_{s = {t_0}}^{t - 1 + \alpha } {{{(t - s - 1)}^{( - \alpha )}}x(s)} } \right) = 0, & t \in {_{{t_0} + 1 - \alpha }},$$ where Δ α denotes the Riemann-Liouville fractional difference operator of order α , 0 < α ≤ 1, ℕ t 0 +1−α ={ t 0 +1−α t 0 +2−α…}, t 0 > 0 and γ > 0 is a quotient of odd positive integers. We establish some oscillatory criteria for the above equation, using the Riccati transformation and Hardy type inequalities. Examples are provided to illustrate the theoretical results.