Journal of Applied Mathematics and Simulation 
Volume 2 (1989), Issue 2, Pages 101-111
doi:10.1155/S1048953389000080

Sharp conditions for the oscillation of delay difference equations

G. Ladas,1 Ch. G. Philos,2 and Y. G. Sficas2

 1Department of Mathematics, The University of Rhode Island, Kingston 02881, RI, USA
2Department of Mathematics, University of Ioannina, P.O. Box 1186, Ioannina 45110, Greece

Abstract

Suppose that {pn} is a nonnegative sequence of real numbers and let k be a positive integer. We prove that limn→∞inf [1k∑i=n−kn−1pi]>kk(k+1)k+1 is a sufficient condition for the oscillation of all solutions of the delay difference equation An+1−An+pnAn−k=0,   n=0,1,2,…. This result is sharp in that the lower bound kk/(k+1)k+1 in the condition cannot be improved. Some results on difference inequalities and the existence of positive solutions are also presented.