Oscillation Theory of Linear Systems

A function f: +→ strongly oscillates if for every t + such that f(t)≠0 there exists s>t with f(t) f(s)<0. Let T be a strongly continuous semigroup with generator A in a real Banach space X. We say that a point x X strongly oscillates if for every ξ X* the function t→ξ(T(t) x) strongly oscillates. If the spectrum of the generator A of semigroup T has no nonnegative real values then almost all points in X strongly oscillate. In the case when A generates a strongly continuous group under the above assumption all points strongly oscillate. We apply the above results to strong oscillation of solutions of difference, differential and functional differential equations.