Oscillation and Nonoscillation of Even - Order Nonlinear Delay - Differential Equations

for their oscillatory and nonoscillatory nature. In Eqs. (1) and (2) y("(x) = (d%/dx')y{x), i = 1, 2, • • • , 2n; yr(x) = y(x — t(x))] dy/dx and d2y/dx2 will also be denoted by y' and y" respectively. Throughout this paper it will be assumed that p(x), j(x), t{x) are continuous real-valued functions on the real line (—°°, °°); j(x), p(x) and t(x), in addition, are nonnegative, r(x) is bounded and f(x), p(x) eventually become positive to the right of the origin. In regard to the function g we assume the following: (i) g : R2 —* R is continuous, R being the real line, (ii) g(\x, \y) — X2a+1^(x, y) for all real \ ^ 0 and some integer q > 1, (iii) sgn g(x, y) = sgn x, (iv) g(x, y) —* as x, y —* <=° ; g is increasing in both arguments monotonically. Eq. (I) is called oscillatory if every nontrivial solution y(t) £ [<0 , 00) has arbitrarily large zeros; i.e., for every such solution y(t), if y(t,) = 0 then there exists <2 > a > 0. A similar definition holds for eq. (2). All solutions of (1) and (2) considered henceforth are continuous and nontrivial, existing on some halfline [t0 , 0° ). 2. Nonoscillation of Eq. (1). We will need the following lemmas. Lemma 1. (Staikos and Petsoulas [8, p. 697].) If y(t) > 0, y'(t) > 0 and y"(t) < 0 for large t, then lim,_.„ (yT(t)/y(t)) = I. Lemma 2. Suppose J" t2"~1p(t) dt — Let y(t) be a nonoscillatory solution of Eq. (2) such that y(t) <0 for large t. Then