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

where t ≥ t0 ≥ 0, a(t), p(t), q(t) ∈ C ([t0,∞) ;R) and Ψ, k, f ∈ C(R,R). It is also assumed that there are positive constants c, c1, μ and γ such that the following conditions are satisfied: (C1) a(t) > 0 and xf(x) > 0 for all x 6= 0; (C2) 0 < c ≤ Ψ(x) ≤ c1 for all x; (C3) γ > 0 and k(y) ≤ γyk(y) for all y ∈ R; (C4) q(t) ≥ 0, f(x) x ≥ μ > 0 for x 6= 0. We recall that a function x : [t0, t1) → R, t1 > t0 is called a solution of Eq. (1.1) if x(t) satisfies Eq. (1.1) for all t ∈ [t0, t1). In what follows, it will be always assumed that solutions of Eq. (1.1) exist for any t0 ≥ 0. Furthermore, a solution x(t) of Eq. (1.1) is called oscillatory if it has arbitrarily large zeros, otherwise it is called nonoscillatory. Finally, we say that Eq. (1.1) is oscillatory if all its solutions are oscillatory.