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Interval oscillation criteria for second order super-half linear functional differential equations with delay and advanced arguments. (English) Zbl 1180.34070

Summary: Sufficient conditions are established for oscillation of second order super half linear equations containing both delay and advanced arguments of the form

φ α (k(t)x ' (t)) ' +p(t)φ β (x(τ(t)))+q(t)φ γ (x(σ(t)))=e(t),t0,

where φ δ (u)=|u| δ-1 u; α>0, βα, and γα are real numbers; k,p,q,e,τ,σ are continuous real-valued functions; τ(t)t and σ(t)t with lim t τ(t)=. The functions p(t), q(t), and e(t) are allowed to change sign, provided that p(t) and q(t) are nonnegative on a sequence of intervals on which e(t) alternates sign.

As an illustrative example we show that every solution of

φ α (x ' (t)) ' +m 1 sintφ β (x(t-π/5))+m 2 costφ γ (x(t+π/20))=r 0 cos2t

is oscillatory provided that either m 1 or m 2 or r 0 is sufficiently large.

34K11Oscillation theory of functional-differential equations