On small solutions of delay equations in infinite dimensions (Q1128241)

The problem under consideration is the delay differential equation \(\dot u(t) = Lu_t, u(0)=x, u_0 =f\), in the state space \(L^p ([-1,0], X)\), with \(1\leq p<\infty\), \(X\) is a Banach space and \(L\) is a bounded linear operator from \(L([-1,0],X)\) into \(X\), \(u_t (s)=u(t+s)\). It is proved that a mild solution \(u(t)\) is a small solution (i.e. \(\| u(t)\| \) decays to zero faster than any exponential) if and only if the Laplace transform of \(u(t)\) extends to an entire function. The same result holds for the state space \(C([-1,0], X)\).