Oscillation criteria for difference equations with damping terms Samir H. Saker a , Sui Sun Cheng b, * a Department of Mathematics, Mansoura University, Mansoura 35516, Egypt b Department of Mathematics, Tsing Hua University, Hsinchu 30043, Taiwan, ROC Abstract By means of generalized Riccati transformation techniques, we establish some new oscillation criteria for second-order nonlinear difference equations with damping. Ó 2002 Elsevier Inc. All rights reserved. Keywords: Difference equation; Riccati transform; Oscillation 1. Introduction In recent years, the asymptotic behavior of second-order nonlinear differ- ence equations has been the subject of investigations by many authors, see e.g. [1–3,7–20,24–26]. In particular, oscillatory behaviors of second-order nonlinear difference equations of the form Dða n ðDx n Þ c Þþ p n ðDx n Þ c þ q n f ðx nþ1 Þ¼ 0; n ¼ n 0 ; n 0 þ 1; ... ; ð1Þ where c > 0 is quotient of positive odd integers, fa n g 1 n¼n 0 is a positive sequence, fp n g 1 n¼n 0 is a nonnegative sequence and fq n g 1 n¼n 0 is a nonnegative sequence possessing a positive subsequence, are obtained by many authors under various additional conditions such as (H1) a n  p n > 0 for all large n, (H2) f : R ! R such that xf ðxÞ > 0 for x 6¼ 0, * Corresponding author. E-mail address: sscheng@math.nthu.edu.tw (S.S. Cheng). 0096-3003/$ - see front matter Ó 2002 Elsevier Inc. All rights reserved. doi:10.1016/S0096-3003(02)00858-5 Applied Mathematics and Computation 148 (2004) 421–442 www.elsevier.com/locate/amc