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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.

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