An extension of a uniform asymptotic stability theorem by Matrosov (Q1101602)

The purpose of this paper is to extend Theorem 1.2 in the paper by \textit{V. M. Matrosov} [(*) J. Appl. Math. Mech. 26, 1337-1353 (1962; Zbl 0123.054)] and to obtain sufficient conditions for uniform asymptotic stability of solutions of ordinary differential equations. Since several conditions in this paper are incorrect, the author would like to correct them in the following way. The conditions (iii), (iii') and (iii'') should be replaced by the following (iii), (iii') and (iii''), respectively. (iii) The family \({\mathcal U}=\{U(\cdot,u(\cdot))+u(\cdot)\) is any continuous function from \({\mathbb{R}}^+\) into \(B_{H\cdot}\}\) is equicontinuous on \({\mathfrak R}^+\). (iii') The family \({\mathcal V}=\{\dot V_{(1)}(\cdot,u(\cdot):u(\cdot)\) is any continuous function from \({\mathbb{R}}^+\) into \(B_{H\cdot}\}\) is equicontinuous on \({\mathfrak R}^+\). (iii'') Let \(A(t,x)\equiv \dot V_{(1)}(t,x)\). \(\dot A_{(1)}(t,x)\) is bounded on \({\mathfrak R}+;\times B_{H\cdot}\). Corollary 2 includes Theorem 1.2 of (*) as a special case.