Oscillations of retarded differential equations of the neutral and the mixed types (Q1263726)

The aim of this paper is to give some general results for the neutral functional differential equation \[ d/dt[x(t)- g(t,x(\cdot))]+f(t,x(\cdot))=0. \] The author discusses the oscillatory property of the neutral delay differential equation \[ d/dt[x(t)- \sum^{M}_{j=1}a_ j(t)x(t-\delta_ j(t))]=-\sum^{N}_{i=1}b_ j(t)x(t-r_ j(t)), \] the differential equation \(\dot x(t)=-px(t- \tau)+qx(t-\sigma)\) and the integro-differential equation \[ d/dt[x(t)- \int^{t}_{t-r(t)}x(s)d_ sp_ 0(t,s)]=-\int^{t}_{t_{- 1}}x(s)d_ sq_ 0(t,s). \] The results of this paper are generalized versions of \textit{B. R. Hunt} and \textit{J. A. Yorke} [J. Differ. Equations 53, 139-145 (1984; Zbl 0571.34057)], \textit{G. Ladas} and \textit{Y. G. Sficas} [Proc. Int. Conf. Qual. Theory Differ. Equations (Canada; 1984)] and \textit{O. Arino}, \textit{G. Ladas} and \textit{Y. G. Sficas} [SIAM. J. Math. Anal. 18, 64-73 (1987; Zbl 0566.34053)].