Asymptotic stability and integral inequalities for solutions of linear systems on Radon-Nikodym spaces

We consider the mild solution u(f)(center dot, 0) of a well-posed nonhomogeneous Cauchy problem { u(t) = A(t)u(t) + f(t), t >= 0 {u(0) = 0 on a Radon-Nikodym space X, where A(center dot) is a linear operator-valued function. Under certain additional conditions we will prove that if the homogeneous system u(t) = A(t)u(t), t >= 0 is exponentially stable, then for each function f belonging to the Sobolev space W-p1(0)(R+, X), 1 <= p < infinity, the solution u(f)(center dot, 0) lies in the same space. The converse statement is more subtle, but it certainly works in the autonomous case. Integral inequalities of Landau type for the evolution semigroup associated with the system (A(t)) on the space W-p1(0)(R+, X) are also derived.