Oscillation for neutral dynamic functional equations on time scales. (English) Zbl 1060.34038

This paper is concerned with the neutral dynamic equation \[ (x(t)-P(t)x(g(t))^\triangle+Q(t)x(h(t))=0 \] on a time scale {T}. An interval condition is imposed on the neutral delay term \(g\) and on the nonneutral deviating argument term \(h\) that allows \(g(t)\) to be locally nonincreasing. The term \(h(t)\) can be delay, advanced, or \(h(t)-t\) can oscillate about zero. The influence of oscillation of \(P(t)\) is addressed along with a positive number for both regular and nonregular oscillation with respect to \(g(t)\). Results involve integrals of combinations of \(P,\;Q,\;g\) and \(h\) as well as auxiliary functions. Applications to difference equations are given.


34K11 Oscillation theory of functional-differential equations
34K40 Neutral functional-differential equations
39A10 Additive difference equations
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