Characterization of Lyapunov pairs in the nonlinear case and applications

The paper presents a characterization of the Lyapunov pairs for a general initial value problem with a possibly multivalued mm-accretive operator on an arbitrary Banach space by means of the contingent derivative related to the operator. The proof is based on tangency and flow-invariance arguments combined with a priori estimates and approximation. The abstract results are applied to obtain precise a priori estimates for the mild solutions. They readily lead to the existence of global solutions and various controllability properties. Important Lyapunov pairs are pointed out in connection with specific problems.