On oscillation of nonlinear second order dierential equation with damping term Mokhtar Kirane a , Yuri V. Rogovchenko b, * a Universit e de Picardie Jules Verne, 52, Boulevard Saint Andr e, 60000 Beauvais, France b Department of Mathematics, Eastern Mediterranean University, Famagusta, TRNC, Mersin 10, Turkey Abstract We present new oscillation criteria for the second order nonlinear dierential equation with damping. Our theorems are stated in general form; they complement and extend related results known in the literature. The relevance of our results is illustrated with a number of examples. To facilitate computations in these examples, we use Mathematica â , a symbolic computer language. Ó 2001 Elsevier Science Inc. All rights reserved. Keywords: Second order nonlinear dierential equation; Damping term; Oscillation; Integral averaging technique 1. Introduction In this paper we are concerned with the problem of oscillation of the non- linear second order dierential equation with a damping term rtx 0 t 0  ptx 0 t qtf xt  0; t P t 0 ; 1 where t 0 P 0; rt2 C 1 t 0 ; 1; 0; 1; pt2 Ct 0 ; 1; 1; 1; qt2 Ct 0 ; 1; 0; 1 and qt is not identical zero on any interval t  ; 1; t  P t 0 ; f x2 C1; 1; 1; 1: A function x : t 0 ; t 1 ! 1; 1; t 1 > t 0 is called a solution of Eq. (1) if xt satis®es Eq. (1) for all Applied Mathematics and Computation 117 (2001) 177±192 www.elsevier.com/locate/amc * Corresponding author. Tel.: +90-392-630-1010; fax: +90-392-365-1604. E-mail address: yuri.as@mozart.emu.edu.tr (Y.V. Rogovchenko). 0096-3003/01/$ - see front matter Ó 2001 Elsevier Science Inc. All rights reserved. PII:S0096-3003(99)00172-1