Non-Oscillation of half-linear difference equations with asymptotically periodic coefficients

Abstract

We study oscillatory properties of half-linear difference equations with asymptotically periodic coefficients, i.e., coefficients which can be expressed as the sums of periodic sequences and sequences vanishing at infinity. Using a special variation of the discrete Riccati technique, we prove that the non-oscillation of the studied equations can be determined directly from their coefficients. Thus, the studied equations can be widely used as testing equations. Our main result is new even for linear equations with periodic coefficients. This fact is illustrated by simple corollaries and examples at the end of this paper.