OSCILLATION THEOREM FOR SECOND-ORDER DIFFERENCE EQUATIONS

Abstract

Sufficient and necessary conditions are established for the second-order difference equations \[ \Delta (r_{n - 1} \Delta x_{n - 1} ) + p_n x^\gamma _n = 0,n = 1,2, \ldots \] where $\gamma $ is the quotient of odd positive integers. Our results extend the well known oscillation theorem which was proved in [1,JMAA,91:9-29,1983], and answer an open problem in [2] when $r_n=1,\gamma=1$.