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Forced oscillation of nth-order nonlinear differential equations. (English) Zbl 1060.34019

Summary: We establish some new oscillation criteria for forced nonlinear differential equations of the form

x (n) (t)+q(t)|x(t)| λ sgnx(t)=f(t),λ>1,

where the coefficient q(t) is allowed to change its sign and the forcing term f(t) is not required to be the nth derivative of an oscillatory function. We further employ the technique presented here to establish new oscillation criteria for forced nonlinear neutral equations of the form

(x(t)+cx[t-τ]) (n) +a(t)x(t)+b(t)x[t-τ]+q(t)|x(t)| λ sgnx(t)+p(t)|x|[t-τ]| σ sgnx[t-τ]=f(t),

with λ>1 and σ>1.

34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory
34K11Oscillation theory of functional-differential equations
34K40Neutral functional-differential equations