On 2-spheres in 4-manifolds.

Definition 3: X(yO) = -sup {X': u(rOuOy0) exp X' T is bounded for all T, 0 . T< +O}. Type numbers (or their negatives, Lyapunov numbers) are usually defined somewhat more generally than this,2 but only the radial type numbers defined above are needed here. Now. any hypothesis concerning the real parts of the characteristic roots of A when k = 1 can be formulated in terms of the radial type numbers of the limit solutions when k . 1. For it can be shown that when k = 1, the set of radial type numbers of the limit solutions of (3) is precisely the set of real parts of the characteristic roots of A. In particular, Lyapunov's theorem can be generalized. THEOREM. If k > 1 and if the radial type number of every limit solution of (3) is negative, then the trivial solution, x = 0, of (1) is asymptotically stable. The proof consists of showing that the hypothesis implies that the radial type number of every solution of (3) is negative and that this in turn implies that the trivial solution of (2) is asymptotically stable. A theorem of Zubov3 is used to prove this latter assertion. Finally, it is a theorem of Massera4 that if the trivial solution of (2) is asymptotically stable, so is the trivial solution of (1). 3. The results of an earlier paper5 and some remarks made by S. Lefschetz were the stimuli for what has been presented here. It should be noted that the main theorem of reference (5) is a special case of the theorem stated above.