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Oscillation of second-order nonlinear neutral delay dynamic equations on time scales. (English) Zbl 1097.39003

This paper deals with second-order nonlinear neutral delay dynamic equations of the form

(r(t)([y(t)+p(t)y(t-τ)] Δ ) γ ) Δ +f(t,y(t-δ))=0

on time scales 𝕋 satisfying the implication t𝕋t-δ,t-τ𝕋 for all t𝕋, where τ,δ are positive reals. The reviewer wants to point out that in case δ>0 or τ>0 this implies bounded graininess of the time scale 𝕋 under consideration. Moreover, γ1 is an odd integer, the positive coefficient functions r,p are assumed to be rd-continuous where r satisfies some integrability condition, p(t)[0,1) and the nonlinearity f is continuous satisfying some positive feedback property.

The author provides various sufficient oscillation criteria for the above equation, in particular of Philos- and Kamenev-type. From these general results, certain special cases for particular time scales are deduced. Referring to the above remark on bounded graininess, Corollaries 4.5 and 4.6 are only applicable in the delay-free case δ=τ=0. Finally, three examples illustrate the paper.

34K11Oscillation theory of functional-differential equations
34K40Neutral functional-differential equations
39A10Additive difference equations