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We present some criteria for the oscillation of the second order nonlinear differential equation [a(t)ψ(x(t))x'(t)]' + p(t)x'(t) + q(t)f (x(t)) =0, tt 0> 0 with damping where aC 1 ([t 0,∞)) is a nonnegative function, p, q∈ C([t 0,∞)) are allowed to change sign on [t 0,∞), ψ, f∈C(R) with ψ(x) ≠ 0, xf(x)/ψ(x) > 0 for x≠ 0, and ψ, f have continuous derivatives on R{0} with [f(x) / ψ(x)]' ≧ 0 for x≠ 0. This criteria are obtained by using a general class of the parameter functions H(t,s) in the averaging techniques. An essential feature of the proved results is that the assumption of positivity of the function ψ(x) is not required. Consequently, the obtained criteria cover new classes of equations to which known results do not apply.

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Using the integral average method, we give some new oscillation criteria for the second order differential equation with damped term  (a(t)Ψ(x(t))K(x'(t)))'+p(t)K(x'(t))+q(t)f(x(t))=0, t≧t0. These results improve and generalize the oscillation criteria in[1], because they eliminate both the differentiability of p(t) and the sign of p(t), q(t). As a consequence, improvements of Sobol's type oscillation criteria are obtained.

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