On Lipschitz stability for comparison systems of differential equations via limiting equation (Q1774822)

Some concepts introduced by the author are valid in any metric normed space. However, in the article under review, the author is primarily interested in the stability of a system of differential equations in \(\mathbb{R}^n\). A nonempty set \(K\) is a cone in \(\mathbb{R}^n\) if \(\lambda K\subset K\) for any \(\lambda> 0\). The author discusses the class of ordinary differential equations \[ u'(t)= G(t,u), \quad u(t_0)= u_0,\tag{1} \] and wishes to establish the stability of the ``zero solution'', by comparing it with a similar equation \[ u'= g(t,u),\tag{2} \] where this equation is the limiting case of a sequence for functions \(G(t+ n,u)\), as \(n\) becomes very large. Definitions are given without explanations and this paper is difficult to read without the knowledge of previous papers of the author and his collaborator M. M. A. El Sheikh. The following is the main result of this paper. The author compares the stability of the \(G\) and \(g\) systems concluding that uniform \(\phi_0\) Lipschitz stability of (2) of the zero solution implies the same stability of (1). Examples showing importance of \(\phi_0\) Lipschitz stability are absent.