极限方程与F完全稳定性

LIMITING EQUATIONS AND F-TOTOL STABILITIES

本文用极限方程的方法研究微分方程零解的F完全稳定性,在两种不同的拓扑——紧开拓扑和Bohr拓扑的意义下,分别研究了所给微分方程零解的F完全稳定性与其极限方程零解的类似稳定性的之间的关系。所得结果表明,用极限方程的方法研究微分方程解的稳定性质与所选定的拓扑密切相关。

In this paper the author studies F-total stabilities of the null solution of a given differential equation by means of limiting equations. Let f∈C(R^+×W; R^n), where W is an open set in R^n with oW. The notation f∈C_o(x) means that, for any compact subest Kin W, there exists a L(K)>0 such that for all converges to f~* in the compact open topology for some converges to f~* in the Bohr topology for some, and consider and its limiting equations The main results of this paper are as follows.Theorem A: Let The solution(1) is eventually F-uniformly totally asymptotically stable (e. F. u. t. a. s. for short) if and only if the null solution of every equation (2) is F-uniformly totally asymptotically stable (F. u. t. a. s for short), uniformly also with respect to. Theorem B: Suppose. If there exists a (x) such that the null solution of (2) is eventually F-uniformly totally stable (e. F. u. t. s. for short), then the null solution of (1) is e.F.u.t.s. Theorem C: Suppose f(t, 0)=0. If there exists a such that and the null solution of (2) is e. F. u. t. a. s., then the null solution of (1) is e.F.u.t.a: s. Theorem D: Let F(t: 0)=0. If there exists (x) such that the solutions of (2) are uniformly bounded, and if the null solution of (2) is uniformly attractive, then the null solution of (1) is e. F. u. t. s. For globally Ru. t. a. s. properties, the author also gets some analogous'results.