Classification of stability-like concepts and their study using vector Lyapunov functions

In the first section, stability-like definitions for ordinary differential equations are derived from a general qualitative concept. It is shown that the classical definitions of stability in the sense of Lyapunov, and their extensions can easily be deduced from this general formulation. A classification of all the definitions which may be derived is proposed.The second section contains the main results of this paper. It deals with the “comparison method” based upon one of T. Wazewski's theorems on differential inequalities. Several authors have used this method in order to investigate stability-like properties. We display the structure of this method, in order to state and prove some general comparison principles. These apply to the class of concepts considered earlier.In the last section some new results about stability and attractivity of sets are obtained as examples for the comparison principles. A theorem on stability in tube-like domains is proved in order to emphasize the generality and the flexibility of the comparison method.