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(Δαx(t)))+p(t)f(∑s=t0t−1+α(t−s−1)(−α)x(s))=0,t∈𝕅t0+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_t0+1−α={t₀+1−αt₀+2−α…}, t₀ > 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.