Nonoscillation criteria for second-order nonlinear differential equations

Consider the second order nonlinear differential equations with damping term and oscillation’s nature of ( ( ) '( )) ' ( ) '( ) ( ) ( ( )) ( '( )) 0 r t x t p t x t q t f x t k x t    0 t t  (1) to used oscillatory solutions of differential equations ( ( ) '( )) ' ( ) ( ( )) ( '( )) 0 t x t t f x t k x t     (2) where ( ) t  and ( ) t  satisfy conditions given in this work paper. Our results extend and improve some previous oscillation criteria and cover the cases which are not covered by known results. In this paper, by using the generalized Riccati’s technique and positive function H (t, s) of Philo we get a new oscillation and nonoscillation criteria for (1).