On oscillation of differential and difference equations with non-monotone delays

For the delay differential equation x'(t) + p(t)x(h(t)) = 0, p(t) >= 0, h(t) <= t, t >= 0, lim(t ->infinity)h(t) = infinity, we demonstrate that the inequality lim sup(t ->infinity) integral(t)(h(t))p(u)du > 1 is not sufficient for oscillation. Moreover, for any A > 0 the relation lim sup(t ->infinity) integral(t)(h(t))p(u)du > A, generally, does not imply oscillation. A similar result is obtained for difference equations. In terms of the maximal argument, new sufficient oscillation conditions are presented for differential and difference equations. (C) 2011 Elsevier Inc. All rights reserved.