EXPONENTIAL STABILITY FOR NONAUTONOMOUS FUNCTIONAL DIFFERENTIAL EQUATIONS WITH STATE-DEPENDENT DELAY

The properties of stability of a compact set K which is positively invariant for a semiflow (Omega x W-1,W-infinity ([-r, 0], R-n), Pi, R+) determined by a family of nonautonomous FDEs with state-dependent delay taking values in [0, r] are analyzed. The solutions of the variational equation through the orbits of K induce linear skew-product semiflows on the bundles K x W-1,W-infinity ([-r, 0]; R-n) and K x C ([-r, 0], R-n). The coincidence of the upper-Lyapunov exponents for both semiflows is checked, and it is a fundamental tool to prove that the strictly negative character of this upper-Lyapunov exponent is equivalent to the exponential stability of K in Omega x W-1,W-infinity ([-r, 0]; R-n) and also to the exponential stability of this compact set when the supremum norm is taken in W-1,W-infinity ([-r, 0]; R-n). In particular, the existence of a uniformly exponentially stable solution of a uniformly almost periodic FDE ensures the existence of exponentially stable almost periodic solutions.