Oscillation in First Order Nonlinear Retarded Argument Differential Equations

A result of Ladas; Lakshmikantham, and Papadakis [1] concerning oscillation caused by lag in linear first order retarded argument differential equations is generalized to the sublinear case. Examples showing that such generalization to the superlinear case is impossible are given. 0. Introduction. It is known ([1], for example) that (A) g e C(R, R), g(t) < t for t e R, g(t) is strictly increasing on R and lim g(t)= + oo, t—>-+ co, (B) a(t) locally integrable, a(t)=0 a.e., and (C) lim sup,^ §lw a(s)ds>l, together imply that every solution to (1) x'(t) + a(t)x(g(t)) = 0 is oscillatory, i.e. has arbitrarily large zeros. We show in this note that this result can be generalized to the sublinear case but a corresponding generalization to the superlinear case fails. In particular we consider the more general retarded argument differential equation (2) x'(t) + a(t)f(x(g(t))) = 0 for t e [b, + oo) where (D) xf(x)>0 for X9*0,feC(R, R), f is nondecreasing with |/(x)|->+ 00 as |x|—>-+co. We also assume g(t) satisfies (A) and a(t) satisfies (B). We shall call / generalized sublinear in case limx^0(x/f(x)) = M< + co, for some M. This includes the sublinear case (see [2]) f(x)=x", 0<<x<l, as well as the linear case. Similarly, / is generalized superlinear in case limx^0(x/f(x))= + oo. For simplicity we drop the word "generalized". 1. Sublinear case. We begin with a lemma. Lemma 1.1. For g satisfying (A) and {tn} defined by t0e R arbitrary and ti+^g-^tA, tn-++ co as h^oo. Received by the editors October 10, 1972 and, in revised form, March 22, 1973. AMS (MOS) subject classifications (1970). Primary 34K15.