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Zbl 1060.34038
Mathsen, Ronald M.; Wang, Qiru; Wu, Hongwu
Oscillation for neutral dynamic functional equations on time scales. (English)
[J] J. Difference Equ. Appl. 10, No. 7, 651-659 (2004). ISSN 1023-6198

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.
[Mingshu Peng (St. John's)]

MSC 2000:: *34K11 Oscillation theory of functional-differential equations
34K40 Neutral equations
39A10 Difference equations

Keywords: time scale; $\Delta$-derivative; neutral functional equation; oscillation

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