Interaction between oscillations and global asymptotic stability in delay differential equations (Q802842)

The author establishes some interesting results concerning the interaction between oscillations and asymptotic stability in the following differential equation with several delays, \(\dot x(t)=\sum^{m}_{k=0}A_ kx(t-\tau_ k)\), where each \(A_ k\) is an \(n\times n\) matrix and \(\tau_ k\) a nonnegative constant. The author generalizes some results of Driver, Rybov and himself on the asymptotic representation of solutions and on nonoscillatory solutions of linear delay equations with small delays, and then applies such a generalization to obtain some nice necessary and sufficient condition for the asymptotic stability of the zero solution to some linear delay equations with several delays. In particular, the author proves that for \(\dot x(t)=- Dx(t-\tau)+Bx(t-\sigma)\), where \(D=diag(d_ 1,...,d_ n)\), \(0<d_ i\tau <1/e\) for \(1\leq i\leq n\), and B is an \(n\times n\) nonnegative matrix, the trivial solution is asymptotically stable iff \(-D+B\) is a nonsingular matrix. Extensions to some perturbed linear systems are also discussed.