Necessary and sufficient stability conditions in an asymptotically ordinary delay differential equation (Q1802871)

Consider the system \(x'(t)=Ax(t-r(t))\); assume that \(\| A\| \limsup_{t\to\infty} r(t)<{1\over e}\), \(\int_ 0^ \infty | r(t)- r_ 0| dt<\infty\), \(r_ 0\geq 0\). Associate the system \(y'(t)=Ay(t- r_ 0)\). Then equi-stability and equi-asymptotic stability respectively, are equivalent for the given system and for the associated one. Under some other assumptions, the same result holds for the associated system \(y'=(A-A^ 2 r(t))y\). The proofs are based on some developments of the fact pointed out by Ryabov and Kurzweil that if the delay is small enough, then the infinite-dimensional space of solutions of the delay- equation is asymptotically equivalent to a finite-dimensional space [see e.g., \textit{J. Kurzweil} and \textit{J. Jarnik}, Boll. Unione Mat. Ital., IV. Ser. 11, Suppl. Fascic. 3, 198-208 (1975)].