STABILITY AND ASYMPTOTIC STABILITY OF FUNCTIONAL-DIFFERENTIAL EQUATIONS

We investigate asymptotic behaviour of solutions of the functional-dif ferential equation y'(t)=f(t,y(t))+g(t,y(theta(t)))+C(t)y'(theta(t)), where f and g are locally Lipschitz functions, C is a continuous matri x and the smooth lag function theta obeys 0 less than or equal to thet a(t) less than or equal to t for t greater than or equal to 0. We tran sform the equation into a delay equation with an infinity of delays an d use a theorem of Soderlind to derive sufficient conditions for stabi lity and for asymptotic stability in the case lim(t-->infinity) theta( t) = infinity. The situation is qualitatively different when lim(t-->i nfinity) theta(t) = theta < infinity and we outline stability conditi ons for that case by employing direct techniques.