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

For the delay differential equationx′(t)+p(t)x(h(t))=0,p(t)⩾0,h(t)⩽t,t⩾0,limt→∞h(t)=∞,we demonstrate that the inequality limsupt→∞∫h(t)tp(u)du>1 is not sufficient for oscillation. Moreover, for any A > 0 the relation limsupt→∞∫h(t)tp(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.