On the asymptotic behavior of the solutions of a class of scalar neutral equations generating a monotone semi-flow (Q923209)

Consider the scalar neutral type equation \[ (1)\quad \frac{d}{dt}\{x(t)- cx(t-1)\}=f(t-1,x(t-1))-f(t,x(t)), \] where f is increasing in x and \(0<c<1\). Using the theory of monotone semi-flows which have a monotone first integral, the authors show that the asymptotic behaviour of bounded solutions depends only on the value of a first integral of (1). If f(t,x) is an almost periodic or periodic function, or even constancy in time, then the positively bounded solutions share asymptotically the same properties. The autonomous and nonautonomous cases, where all the solutions are bounded are also considered.