OSCILLATION CRITERIA FOR ODD ORDER NEUTRAL EQUATIONS

[摘要] In this paper four main results are obtained for oscillation of all solutions of the odd order neutral differential equation [GRAPHICS] where p(i)(t) greater than or equal to sigma(i), is an element of (0, infinity), f(i): R --> R (i = 1, 2, ..., m), and p, tau is an element of [0, infinity). Theorem 1 shows that if, in addition to the above, 0 less than or equal to p < 1, f(i)(x) = x, n > 1, and, for some mu is an element of (0, 1), all solutions of the first order delay equation [GRAPHICS] are oscillatory, then all solutions of (*) are oscillatory. In particular, when m = 1 and p(1)(t) = p(1) is an element of (0, infinity) then p(1) sigma(1)(n) > ((1 - p)(n)!/e)(n/(n - 1))n(-1) implies that all solutions of(*) are oscillatory. In such case Theorem 4.1, due to Gopalsamy, et al. [Czech. Math. J. 42 (1992), 313-323] reduces to p(1) sigma(1)(n) > (1 - p)(n) ! which, in view of the known inequality (1/e)(n/(n - 1))n(-1) < 1, is a stronger condition. Some results of Zhang V. Math. Anal. Appl. 139 (1989), 311-318], Graef ct al. [J. Math Anal. Appl. 155 (1991), 562-571], and Ladas et al. [Proc. Amer. Math. Sec. 113 (1991), 123-133] have been generalized in Theorems 2, 3, and 4, respectively. (C) 1994 Academic Press, Inc.

[发布日期] 1994-11-15