Oscillation Criteria for a Class of Second-Order Nonlinear Differential Equations with Damping Term

and Applied Analysis 3 Proof. Assume that 1.1 has a nonoscillatory solution x t . Without loss of generality, suppose that it is an eventually positive solution if it is an eventually negative solution, the proof is similar , that is, x t > 0 for all t ≥ t0. We consider the following three cases. Case 1. Suppose that x′ t is oscillatory. Then there exists t1 ≥ t0 such that x′ t1 0. From 1.1 , we have ( r t ∣∣x′ t ∣∣σ−1x′ t exp (∫ t t0 p s r s ds ))′ ( r t ∣∣x′ t ∣∣σ−1x′ t )′ exp (∫ t t0 p s r s ds ) p t ∣∣x′ t ∣∣σ−1x′ t exp (∫ t t0 p s r s ds ) −q t f x t exp (∫ t t0 p s r s ds ) < 0, 2.5 which means that r t ∣∣x′ t ∣∣σ−1x′ t exp (∫ t t0 p s r s ds ) < r t1 ∣∣x′ t1 ∣∣σ−1x′ t1 exp (∫ t1 t0 p s r s ds ) 0, t > t1, 2.6 it follows that x′ t < 0 for all t > t1, which contradicts to the assumption that x′ t is oscillatory. Case 2. Suppose that x′ t < 0. From 1.1 , we obtain − ( r t ∣∣x′ t ∣∣σ−1x′ t )′ ( r t (−x′ t )σ)′ −p t (−x′ t )σ q t f x t ≥ 0, 2.7 then there exists anM > 0 and a t1 ≥ t0, such that r t (−x′ t )σ ≥ M, t ≥ t1, 2.8