ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF DIFFERENTIAL EQUATIONS WITH VARIABLE DELAYS

The authors consider the system of forced differential equations with variable delays $$x'(t) + \sum^N_{j=1}B_j(t)x(t-\tau_j(t)) = F(t)\eqno(*)$$ where $B_j(t)$ is a continuous $n\times n$ matrix on ${{\Bbb R}^+}$, $F\in C({{\Bbb R}^+, {\Bbb R}^n})$ and $\tau \in C({{\Bbb R}^+, {\Bbb R}^+})$. Using Razumikhin-type techniques and Liapunov's direct method, they establish conditions to ensure the ultimate boundedness and the global attractivity of solutions of $(*)$, and when $F(t) \equiv 0$, the asymptotic stability of the zero solution. Under those same conditions, they also show that $\int^{+\infty}_0\sum_{j=1}^{N}|B_j(t)|\,dt = +\infty$ is a necessary and sufficient condition for all of the above properties to hold. 1991 Mathematics Subject Classification: 34K15, 34C10.