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Oscillation criteria for second-order retarded differential equations. (English) Zbl 0902.34061

The authors investigate oscillatory properties of second-order quasilinear equations

[r(t)|u ' (t)| α-1 u ' (t)+p(t)|u(τ(t))| β-1 u(τ(t))]=0(*)

with α,β>0, r(t)>0, p(t)0, τ(t)t and lim t τ(t)=. The results deal mostly with the case α=β and extend some earlier criteria for linear equations u '' +p(τ(t))=0 given by L. Erbe [Canadian Math. Bull. 16, 49-56 (1973; Zbl 0272.34095)] and J. Ohriska [Czech. Math. J. 34, 107-112 (1984; Zbl 0543.34054)]. A typical result is the following oscillation criterion:

Equation (*) with α=β and r(t)1 is oscillatory provided one of the following conditions holds:

lim t t α t p(s)τ(s) s α ds>1,orlim sup t t α γ(t) p(s)ds>1,

with γ(t)=sup{s:τ(s)t}.

Reviewer: O.Došlý (Brno)
34K11Oscillation theory of functional-differential equations
34C15Nonlinear oscillations, coupled oscillators (ODE)