Asymptotic Periodicity, Monotonicity, and Oscillation of Solutions of Scalar Neutral Functional Differential Equations

We consider the periodic scalar neutral functional differential equation drdt.wx t.yc t.x tyt .xsyh t, x t..qh tys, x tys.., where c is continuously differentiable, h is increasing in its second argument, and both c and h are 1-periodic in the t-variable. The two time-lags t and s are not required to be the same. It is shown that, under certain conditions, i. the set of 1-periodic solutions is an ordered arc and each solution is convergent to a periodic solution; ii. the asymptotic and oscillatory behaviors of each solution are completely classified in terms of the value of the first integral at the initial condition