On nonlinear oscillations for a second order delay equation

Oscillation, continuation, and uniqueness of solutions of retarded differential equations

“…Atkinson [1] gave necessary and sufficient conditions for all solutions of (N) to oscillate when a(t) > 0 and f(x) = x2""1"1 for « > 1. That result was extended by Gollwitzer [11] to (D) x" + a(t)f(x(t -r(t))) = 0 for a(t) > 0, t(/) positive and bounded, and/(x) = xy where y is the ratio of odd positive integers and y^l, An example by Waltman [16] can be modified to show that Gollwitzer's result will fail if r(t) is allowed to become unbounded.…”

“…Notice also that the remaining theorems differ significantly from many results of previous investigators (cf. [11], [2], and [15]) in that r(t) need not be bounded. A notable exception to this is found in the work of Eliason [10] whose specific goal was to obtain the results of the previous investigators of the problem with f(x) = |x|rsgn x without assuming r(t) bounded.…”

Strong oscillations for second order nonlinear functional differential equations

“…Equation (I.I) is said to be oscillatory if every solution of (I.I) is oscillatory Oscillation theory for (I.I) has been developed by many authors. Bradley [I], Chiou [2], Erbe [3] Gollwitzer [4], Ladas [5], Travis [6], Waltman [7], Wong [8] and references therein It is wellknown theorem of Wintner [9] and Leighton [I0] It is easy to verify that x"(t) < 0 and x'(t) >0 for all large t. Let…”

Oscillation Criteria and Growth of Nonoscillatory Solutions of Even Order Ordinary and Delay-Differential Equations

“…For results on oscillation of solutions of delay-differential equations we refer the reader to Burton and Grimmer [1], [2], Erbe [4], Gollwitzer [5], and Wong [19] for the second-order case and Ladas [12] for the higher order case.…”