On Global Asymptotic Stability of Solutions of Differential Equations(1)

I ■V *■ • • • iV*TC | V j j J j (1.1) x'=/(x) in which f(x) is of class C1 on En. Let J(x) = (df/dx) denote the Jacobian matrix of /and let H(x) = (J+J*)/2 be the symmetric part of J(x). One of the results of [2] is to the effect that if (1.2) /(0) = 0 and (1.3) H(x) is negative definite (for fixed x ^ 0), then x = 0 is a globally asymptotically stable solution of (1.1); i.e., every solution x = x(t) or (1.1) exists for large t and x(t)—>0 as t—»oo. Among the results of [4], which deals with the case w = 2, is the following: if (1.4) x = 0 is a locally asymptotically stable solution of (1.1) and (1.3) is replaced by the conditions (1.5) tr/(x) = tx H(x) ^ 0, (1.6) |/(x)| ^ Const. > 0 for |x| è const. > 0, then again x = 0 is a globally asymptotically stable solution of (1.1). One of the main results of the first part of this paper will be a generalization of the latter theorem to the case of arbitrary n ^2. In this situation, the trace of H(x) will be replaced by the function (1.7) a(x) = max(X¿(x) + Xj(x)) for 1 ^ i < / ^ n, where Xi(x), • • • , X„(x) are the eigenvalues of H(x), the condition (1.5) by (1.8) a(x) ^ 0,