Oscillation Tests: Second-order Neutral Difference Equation

In this work, we present some new sufficient conditions for the oscillation of solutions to the second-order difference equation Δ (r(n)(Δz(n))γ) + Σmi-1 qi(n)xαi(δi(n)) = 0, n ≥ n0 where z(n) = x(n)+ p(n)x(τ (n)) and ∆x(n) = x(n + 1) −x(n). Under the assumption ∞ r(s) −1/α = ∞, we consider two cases when γ ≥ αi for at leat one index i, and general case for αi and γ. Also, we state an open problem.