ASYMPTOTIC PERIODICITY, MONOTONICITY, AND OSCILLATION OF SOLUTIONS OFSCALAR NEUTRAL FUNCTIONAL-DIFFERENTIAL EQUATIONS

We consider the periodic scalar neutral functional differential equati on (d/dt)[x(t) - c(t)x(t - tau)] = -h(t, x(t)) + h(t - sigma, x(t - si gma)), where c is continuously differentiable, h is increasing in its second argument, and both c and h are 1-periodic in the t-variable. Th e two time-lags tau and sigma are not required to be the same. It is s hown that, under certain conditions, (i) the set of 1-periodic solutio ns is an ordered are and each solution is convergent to a periodic sol ution; (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. (C) 1996 Academic Press, Inc.