comparison results and linearized oscillations for higher-order difference equations

Consider the difference equationsΔmxn+(−1)m+1pnf(xn−k)=0,   n=0,1,…        (1)andΔmyn+(−1)m+1qng(yn−ℓ)=0,   n=0,1,….       (2)We establish a comparison result according to which, when m is odd, every solution of Eq.(1) oscillates provided that every solution of Eq.(2) oscillates and, when m is even, every bounded solution of Eq.(1) oscillates provided that every bounded solution of Eq.(2) oscillates. We also establish a linearized oscillation theorem according to which, when m is odd, every solution of Eq.(1) oscillates if and only if every solution of an associated linear equationΔmzn+(−1)m+1pzn−k=0,   n=0,1,…         (*)oscillates and, when m is even, every bounded solution of Eq.(1) oscillates if and only if every bounded solution of (*) oscillates.