On Liapounoff's Conditions of Stability

In his well known paper, Problbme general de la stabilite du mouvement 3, Liapounoff has given several general criteria for the stability of a solution of a system of differential equations, independently of the consideration of the variational equations. These criteria have been further generalized by several Soviet mathematicians (see References at the end of this note). In the present paper, I introduce a new and more strict type of stability, equiasymptotic stability, whose relations to the asymptotic stability of Liapounoff are discussed (Theorem 7 and Example 4). I will show that the sufficient conditions assumed in a theorem of Liapounoff on asymptotic stability and in a theorem of Malkin on simple stability, imply even equiasymptotic stability (Theorem 5). On the other hand, I show that the above mentioned theorem of Liapounoff can be proved under weaker assumptions (Theorem 6). This weakening of the sufficient conditions is needed in order to prove the reciprocal theorem, which is not true, even in very simple cases, under the more restrictive conditions of Liapounoff. Necessary and sufficient conditions for asymptotic stability are thus found in the case of periodic (Theorem 8) or linear (Theorem 9) systems. 2. DEFINITIONS AND NOTATIONS. Consider a system