Sufficient Conditions for the Oscillation of Delay Difference Equations CH.G. PHILOS*, I.K. PURNARAS † and I.P. STAVROULAKIS ‡ Department of Mathematics, University of Ioannina, P.O. Box 1186, 451 10 Ioannina, Greece (Received 17 October 2003; In final form 10 November 2003) The most important result of this paper is a new oscillation criterion for delay difference equations. This criterion constitutes a substantial improvement of the one by Ladas et al. [J. Appl. Math. Simulation 2 (1989), 101 – 111] and should be looked upon as the discrete analogue of a well-known oscillation criterion for delay differential equations. Keywords: Delay difference equation; Solution; Oscillation; Nonoscillation 2000 Mathematics Subject Classification: 39A10; 39A11 INTRODUCTION In the last two decades, the study of difference equations has been the focus of great attention by many researchers. Besides its mathematical interest, the theory of difference equations is also very interesting because of the fact that difference equations arise in various fields of applied sciences more frequently than ever. In particular, the study of the oscillation of solutions of difference equations has attracted a lot of activity. It is the main purpose of this paper to establish a new oscillation criterion for linear delay difference equations with variable coefficients. This criterion substantially improves the one by Ladas et al. [16] and should be looked upon as the discrete analogue of a well-known integral oscillation result [11,12] for first order linear delay differential equations with variable coefficients. Consider the delay difference equation x nþ1 2 x n þ p n x n2k ¼ 0; ðEÞ where ð p n Þ n$0 is a sequence of nonnegative real numbers and k is a positive integer. By a solution of (E), we mean a sequence ðx n Þ n$2k of real numbers which satisfies (E) for all n $ 0: A solution ðx n Þ n$2k of (E) is said to be oscillatory if the terms x n of the sequence are neither eventually positive nor negative, otherwise the solution is called nonoscillatory. Journal of Difference Equations and Applications ISSN 1023-6198 print/ISSN 1563-5120 online q 2004 Taylor & Francis Ltd http://www.tandf.co.uk/journals DOI: 10.1080/10236190410001648239 *Corresponding author. sE-mail: cphilos@cc.uoi.gr † E-mail: ipurnara@cc.uoi.gr ‡ E-mail: ipstav@cc.uoi.gr Journal of Difference Equations and Applications, Vol. 10, No. 4, 10 April 2004, pp. 419–435