Asymptotic convergence criteria of solutions of delayed functional differential equations

The convergence of all solutions of the scalar equation ˙x = f (t,xt ) is studied under the main assumption that every constant is its solution. The criterion and the sufficient conditions for convergence are proved. For the considered classes of equations it is, among others, proved that all solutions are convergent if and only if there exists at least one increasing and convergent solution. As applications, criteria for some classes of linear equations as well as for nonlinear equations are obtained. Some comparisons with already known results are given too.  2002 Elsevier Science (USA). All rights reserved.