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Oscillation criteria for sublinear differential equations with damping

Abstract

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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Manojlovič, J.V. Oscillation criteria for sublinear differential equations with damping. Acta Mathematica Hungarica 104, 153–169 (2004). https://doi.org/10.1023/B:AMHU.0000034369.84782.0a

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  • oscillation
  • sublinear differential equation
  • damping term
  • integral averages