NECESSARY AND SUFFICIENT CONDITIONS FOR EVENTUALLY VANISHING OSCILLATORY SOLUTIONS OF FUNCTIONAL EQUATIONS WITH SMALL DELAYS

Necessary and sufficient conditions are found for all oscillatory solutions of the equation (rn_l(t) (rn_2(t) (---(r2(t) (rl(t)y’ (t)) ’) ’) ’---) + a(t)h(y(g(t))) b(t) to approach zero. Sufficient conditions are also given to ensure that all solutions of this equation are unbounded.

B. SINGH conditions ensured that all nonoscillatory solutions of (I) approached zero.
Kartsatos [4,Theo. I] also found sufficient criteria for all bounded nonoscillatory solutions of (i) to, asymptotically, vanish generalizing results of this author and Dahiya [7,Theo. I]. In fact, since the work of Hammett [3] such asymptotic results about the nonoscillatory solutions of ordinary and retarded differential equations have been obtained by many authors such as Kartsatos [4], Kusano and Onose [5,6], this author and Dahiya [7], this author [9, I0, ii] and many others. A fairly exhaustive list of references on oscillation can be found in Graef [2]. Most of these results relate to nonoscillation properties of solutions. Very little has been said about the asymptotic nature of the corresponding oscillatory solutions of these equations.
This author's work [8,9,12] is devoted to this type of study about the oscillatory solutions of such equations.
Our purpose.in this paper is to further the study initiated by Kusano and Onose [6] and find necessary and sufficient conditions to ensure that all oscillatory solutions of equation (I) tend to zero as t / . In the last section, we give sufficient conditions which cause all solutions of (i) to be unbounded, then [i] studied a similar problem but our results are different and more extensive.
In what follows, we shall restrict our study to those solutions of (I) which can be continuously extended on some positive half line, say for t _> t O > o. We shall, therefore, assume the point t O fixed for the rest of this paper. The term "solution" applies only to continuously extendable solutions on R + [to, ).
The following conditions hold for the rest of this paper: (ii) g(t) < t, g(t) + as t / ; (iii) t h(t) > o, t + o; (iv) there exists a number m such that h(t___) < m A solution is said to be oscillatory if it has arbitrarily large zeros, otherwise it is said to be nonoscillatory. To further shorten notations we designate: ZlY(t (rl(t)y,(t)), Z2Y(t) (r2(t) (rl(t)y,(t)),), 3.
MAIN RESULTS.
PROOF. Since y(t) is oscillatory, ZiY(t) is oscillatory for i I, Due to conditions on r I (t) we find that each term on the right hand side of (13) except possibly last two are bounded, and since h(t)/t < m, there exist constants K I, K 2 and K 3 such that ft sX (X-s)n-2 Rearranging constants still further we get S t S x la(s) llY(g(s))l dsdx IY(g(t)) < K 1 + K 4 t0 t0  (14) and (15), there exists a positive constant K 6 such that By Gronwall's inequality y(g (t)) is bounded and the proof is complete. Xn-la x I x 2 x n + tl/rl(x I) I I/r 2(x 2) i/r 3(x3) f ib(x)dxdXn_l dx I.
This shows that Zn_lY(t is eventually positive contradicting the fact that y (t) is oscillatory.
REMARK. It is to be noted that the conditions rl(t) tn_ ;a(t)dt < and Ib(t) Idt <are not needed here. PROOF. Due to Theorem 3.6 we only need to prove it for a nonoscillatory solution. Let y(t) be nonoscillatory and bounded. From inequality (34) in the proof of Theorem 3.6 it follows that IZn_2Y(t) / as t / . Since Zn_2Y (rn_2Zn_3Y(t))' and rn_ 2 is bounded, we have Zn_3Y(t) / -+ .
Proceeding this way we find that y' (t) / + forcing y(t) / -+ . The proof is now complete by contradiction.
The restriction on r(t) cannot be weakened i.e. 8 cannot be greater than or equal to 1 as the following example shows.