Pergamon Computers Math. Applic. Vol. 29, No. 7, pp. 83-88, 1995 Copyright©1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0898-1221/95 $9.50 + 0.00 0898-1221(95)00020-8 Oscillations of Delay Difference Equations I. P. STAVROULAKIS Department of Mathematics, University of Ioannina 451 10 Ioannina, Greece ipstav@cc, uoi. gr (Received December 1993; accepted February 1994) Abstract--Consider the delay difference equation Xn+l-xn+PnXn-k=O, n-- 0,1,2,..., where {Pn} is a sequence of real numbers and k is a positive integer. New oscillation criteria for this equation are established. An extension of the results to difference equations with several delays is also presented. 1. INTRODUCTION Recently there has been an increasing interest in the study of the oscillatory and asymptotic behavior of the solutions of the delay difference equation Axn + pnxn-k = O, n = O, 1,2,... (*) where {Pn} is a sequence of real numbers, k is a positive integer, and A denotes the forward difference operator Axn = Xn+l - xn. See, for example, [1-5] and the references cited therein. Throughout this paper, the sequence {Pn} is supposed to be defined for n _> 0. By a solution of equation (.) we mean a sequence {x,~} which is defined for n > -k and which satisfies equation (1) for n > 0. A solution {xn} of equation (*), is said to be oscillatory if the terms xn of the solution are neither eventually all positive nor eventually all negative. Otherwise, the solution is called nonosciUatory. Erbe and Zhang [1] proved that, if Pn >- O, then either one of the following conditions k k lira inf Pn > (61) n-~oo (k + 1) k+l or n limsup Z Pi>l (62) rt --* OO i=n--k implies that all solutions of equation (*) oscillate. Then Ladas, Philos and Sficas [5] proved that the same conclusion holds if Pn >_ 0 and lim_ inf Z Pi > (k + 1) k+l" (63) i=n - - k The author would like to thank Y. G. Sficas and Y. Domshlak for their helpful suggestions. This manuscript was originally submitted and accepted for publication in a special issue of this journal, guest edited by R. P. Agarwai, entitled, "Advances in Difference Equations." However, due to last minute revisions by the author, the manuscript was withdrawn from the special issue. Typeset by AA,~%TEX 83