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

AbstractWe consider the periodic scalar neutral functional differential equation (d/dt)[x(t)−c(t)x(t−τ)]=−h(t,x(t))+h(t−σ,x(t−σ)), wherecis continuously differentiable,his increasing in its second argument, and bothcandhare 1-periodic in thet-variable. The two time-lags τ and σ 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