On second order functional differential equations and inequalities with deviating arguments

A necessary and sufficient condition is established in order that (i) the retarded differential equation $$y''(t) = p_0 y(t) + f(y(t - tau _1 ),...,y(t - tau _N ))$$ has no bounded nonoscillatory solution and (ii) the advanced differential equation $$y''(t) = p_0 y(t) + f(y(t + tau _1 ),...,y(t + tau _N ))$$ has no unbounded nonoscillatory solution, wherep ₀≥0 and τ _j > 0,1 ?i ?N, are constants. Differential inequalities related to (^*) and (^**) are also studied. Finally, an oscillation criterion is given for a class of differential equations containing both retarded and advanced arguments.