Comparison results for functional differential equations with two middle terms (Q1070102)

Consider the delay functional differential equations with two middle terms \[ (I)\quad L(x)\equiv x^{(n)}+p(t)x^{(n-1)}+q(t)x^{(n- 2)}+H(t,x(g(t)))=Q(t) \] and (II) \(L(x)=0\). In this paper following comparison results for (I) are presented: 1) The existence of a positive solution to the inequality \(L(x)<0\) implies the same fact for (II); 2) (I) is ocillatory if and only if (II) oscillates under certain conditions; 3) The equation \[ x^{(n)}+p_ 2(t)x^{(n-1)}+q_ 2(t)x^{(n-2)}+H_ 2(t,x(g_ 2(t)))=0 \] is oscillatory if \[ x^{(n)}+p_ 1(t)x^{(n-1)}+q_ 1(t)x^{(n-2)}+H_ 1(t,x(g_ 1(t)))=0 \] is oscillatory under certain conditions; 4) The equation \[ x^{(n)}+p_ 2(t)x^{(n-1)}+q_ 2(t)x^{(n-2)}+H_ 2(t,x(g_ 2(t)))=Q(t) \] is oscillatory, if \[ x^{(n)}+p_ 1(t)x^{(n-1)}+q_ 1(t)x^{(n-2)}+H_ 1(t,x(g_ 1(t)))=Q(t) \] under certain conditions. 1) extends a result of \textit{W. A. Kosmala} [Nonlinear Anal., Theory Methods Appl. 6, 1115-1133 (1982; Zbl 0496.34019)]. 2) extends a result of \textit{A. G. Kartsatos} and \textit{J. Toro} [SIAM J. Math. Anal. 10, 86-95 (1979; Zbl 0402.34043)]. 3) and 4) extend results of \textit{K. E. Foster} and \textit{R. C. Grimmer} [J. Math. Anal. Appl. 77, 150-164 (1980; Zbl 0455.34053)]; \textit{A. G. Kartsatos} [J. Math. Anal. Appl. 52, 1-9 (1975; Zbl 0327.34012)]; \textit{A. G. Kartsatos} and \textit{H. Onose} [Bull. Aust. Math. Soc. 14, 343-347 (1976; Zbl 0318.34044)]. Finally, two oscillation results are given for (I).