Oscillations of delay difference equations in a critical state

AbstractConsider the first-order delay difference equationΔxn + ∑i=1m Pi(n)xn−ki = 0, n = 0,1,2,…,in the case where lim infn→∞ Pi(n) = pi ≥ 0, ki > 0, i = 1,2,…. A necessary and sufficient condition for the oscillation of all solutions of the above equation is established in the critical state that the corresponding “limiting” equationΔxn + ∑i=1m Pixn−ki = 0, n = 0,1,2,…,admits a nonoscillatory solution. It is to be pointed out, that there is no result in this critical state for a delay difference equation with more than one delay argument