## Asymptotic behaviour of functional-differential equations with proportional time delays.(English)Zbl 0856.34078

The problem $y'(t)= Ay(t)+ \sum^\infty_{i= 1} B_i y(q_i t)+ \sum^\infty_{i= 1} C_i y'(p_i t),\quad t> 0,$ $$y(0)= y_0$$, $$p_i, q_i\in (0, 1)$$, is considered. References are indicated for applications, motivations and special cases previously studied. It is proved that every solution is $$o(e^{\alpha t})$$ as $$t\to \infty$$ for any fixed constant $$\alpha> \max\{\alpha(A), 0\}$$, where $$\alpha(\cdot)$$ denotes the maximal real part of the eigenvalues of its argument. Detailed analysis is performed for the cases $$\alpha(A)> 0$$, $$\alpha(A)= 0$$, $$\alpha(A)< 0$$.

### MSC:

 34K99 Functional-differential equations (including equations with delayed, advanced or state-dependent argument) 34K40 Neutral functional-differential equations 34K20 Stability theory of functional-differential equations 34K25 Asymptotic theory of functional-differential equations
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### References:

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