Asymptotic periodicity, monotonicity, and oscillation of solutions of scalar neutral functional differential equations (Q1916841)

Periodic solutions to the scalar neutral functional-differential equation (1) \(d/dt [x(t) - c(t) x(t - s)] = - h(t, x(t)) + h(t - s, x(t - s))\) are considered. Both \(c\) and \(h\) are 1-periodic, \(h\) is increasing in \(x\). It is shown that the set of 1-periodic solutions is an ordered arc and each solution is convergent to a periodic solution. The asymptotic behavior of each solution is classified in terms of the value of the first integral at the initial condition. A wide description of earlier results connected with special cases of (1) is considered.