The author considers the th order functional differential equation
where , , , , , , are continuous, , , for and and . It is shown (Lemma 1): If is a nonoscillatory solution of (1) then there exist a and an integer , such that is even and on , , on , . The integer is called the degree of . This result generalizes a well-known lemma of I. T. Kiguradze [On the oscillation of solutions of the equation , Mat. Sb., n. Ser. 65(107), 172-187 (1964; Zbl 0135.143)].
Denote by the set of all nonoscillatory solutions of degree of (1) and by (resp. ) the set of all nonoscillatory solutions of (1) with odd (resp. even ). We say that (1) has property (A) if is odd and and (1) has property (B) if is even and . In the paper, it is next considered a “comparison” equation (2) , where , , . The author proves that (A) property (resp. (B) property) of (2) implies (A) property (resp. (B) property) of (1) provides that some sign conditions among the functions , , and , , , hold. Similar results for (1) with were proved by S. R. Grace and B. S. Lalli [Math. Nachr. 144, 65-79 (1989; Zbl 0714.34106)].