Nonlinear Analysis 41 (2000) 1005 – 1028 www.elsevier.nl/locate/na Oscillation theorems for second-order equations with damping 1 Yuri V. Rogovchenko a;b; * a Institute of Mathematics, National Academy of Sciences, Tereshchenkivs’ka Str. 3, 252601 Kyiv, Ukraine b Department of Mathematics, Eastern Mediterranean University, Famagusta, TRNC, Mersin 10, Turkey Received 5 February 1997; accepted 14 August 1998 Keywords: Second order; Nonlinear dierential equation; Damping; Oscillation 1. Introduction In this paper, we study the problem of oscillation of the nonlinear second-order dierential equation with damping [r (t )x ′ (t )] ′ + p(t )x ′ (t )+ q(t )f(x(t )) = 0; (1) where r (t )∈C 1 ([t 0 ; ∞); (0; ∞));p(t );q(t ) ∈ C ([t 0 ; ∞); (−∞; ∞));f(x) ∈ C ((−∞; ∞); (−∞; ∞)) and xf(x) ¿ 0 for x =0;t 0 ≥ 0. We recall that a function x :[t 0 ;t 1 ) → (−∞; ∞);t 1 ¿t 0 is called a solution of Eq. (1) if x(t ) satises Eq. (1) for all t ∈ [t 0 ;t 1 ). In what follows, it will be al- ways assumed that solutions of Eq. (1) exist for any t 0 ≥ 0. Furthermore, a solution x(t ) of Eq. (1) is called continuable if x(t ) exists for all t ≥ t 0 . A continuable solution x(t ) of Eq. (1) is called oscillatory if it has arbitrarily large zeroes, otherwise it is called nonoscillatory. Finally, it is said that Eq. (1) is oscillatory if all its solutions are oscillatory. * Correspondence address: Department of Mathematics, Eastern Mediterranean University, Famagusta, TRNC, Mersin 10, Turkey. E-mail address: Rogovch y@member.ams.org (Y.V. Rogovchenko) 1 This research was supported by a fellowship of the Italian Consiglio Nazionale delle Ricerche. 0362-546X/00/$ - see front matter ? 2000 Elsevier Science Ltd. All rights reserved. PII: S0362-546X(98)00324-1