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Stability and stability domains analysis for nonlinear differential- difference equations. (English) Zbl 0807.34089
If a time-delay system $\dot x(t)= A(t,x(t),x(t-\tau(t))) x(t)+ B(t,x(t),x(t-\tau(t)) x(t- \tau(t)))\tag{1}$ and a vector norm $$p$$ are given, then an overvaluing system of (1) with respect to $$p$$ and to the given region $$D$$ is introduced with the property if the inequality $$z(t)\geq p(x(t))$$ between the solutions $$x$$ of (1) and $$z$$ of $\dot z(t)= M(t,x(t),x(t-\tau(t)))z(t)+ N(t,x(t),x(t-\tau(t))) z(t- \tau(t))\tag{2}$ holds on the initial set, then it holds as long as $$x(t)$$ remains in $$D$$. Then sufficient conditions are given for the zero solution of (1) to be stable or asymptotically stable.

##### MSC:
 34K20 Stability theory of functional-differential equations 34K25 Asymptotic theory of functional-differential equations 34A40 Differential inequalities involving functions of a single real variable