Asymptotic constancy of solutions of delay-differential equations of implicit type (Q1293397)

The authors study delay differential equations of the form \(x'(t)=f(t,x(t-r(t,x(t)))-x(t))\) and \(x'(t)=g(t,x(t))-g(t,x(t-r(t,x(t))))\) with continuous functions \(f\) and \(g\), satisfying \(|f(t,x)|\leq \mu(t)|x|\), \(|g(t,x)-g(t,y)|\leq\mu(t)|x-y|\), where \(\mu(t)\) is continuous. Using the fixed point theorem of Tikhonov, the authors prove that any solution converges under some additional conditions on the functions \(r\) and \(\mu\). Moreover, for any possible \(\xi\), there exists a solution \(x\) such that \(x(t)\to\xi\). It is shown that these results are also valid, in some cases, for several lags.