Existence and stability of almost periodic solutions for quasilinear delaysystems and the Halanay inequality

We present some easily verifiable conditions for the existence and global a symptotical stability of almost periodic solutions to systems of delay diff erential equations in the form u'(t)= -Au(t) + Wg(t,u(t)) + f(t), where A i s an element of L(R-n) and f(t), g(t, p) are almost periodic in t uniformly on p from bounded subsets of C([-h,0], R-n). With this purpose, we use the exponential dichotomy theory together with usual positivity arguments. In particular, the semigroup version of the so-called Perron-Frobenius theorem is applied to study the generalized Halanay inequality. Finally, systems w ith maxima are studied in detail, (C) 2000 Academic Press.