Stability relative to a set and to the whole space revisited

摘要

In [1] P. Seibert suggested without proof some theorems on stability. These theorems involved two new concepts: The first is a local stability of a subset Y of a metric space X on a subset Z of Y. The second is a notion of conditionally attracting set Y ⊆ X on Z ⊆ Y. Using the first of Seibert's concepts, we suggest new theorems and provide the corresponding proofs. However, our theorems are logically independent of those of Seibert. In addition, a proof of a criterion on local stability suggested by Seibert is given.