*(English)*Zbl 0603.34069

The equations treated in this paper are of the form (*) $P\left(z\right)+Q\left(z\right){e}^{-Tz}=0$ and arise from the consideration of delay-differential equations with a single delay T, $T\ge 0$. The equation is called stable if all zeroes lie in $Re\left(z\right)<0$ and unstable if at least one zero lies in $Re\left(z\right)>0$. A stability switch is said to occur if the equation changes from stable to unstable, or vice-versa, as T varies through particular values. The main result is the following theorem.

Assume that: (i) P(z), Q(z) are analytic functions in $Re\left(z\right)>\delta $ $(\delta >0)$ which have no common imaginary zero; (ii) the conjugates of P(-iy) and Q(-iy) are P(iy) and Q(iy) for real y; (iii) $P\left(0\right)+Q\left(0\right)\ne 0;$ (iv) there are at most a finite number of zeroes of P(z) $+Q\left(z\right)$ in the right half-plane; (v) $F\left(y\right)\equiv {\left|P\left(iy\right)\right|}^{2}-{\left|Q\left(iy\right)\right|}^{2}$ for real y, has at most a finite number of real zeroes.

Under these conditions, the following statements are true: (a) If the equation $F\left(y\right)=0$ has no positive roots, then if (*) is stable [unstable] at $T=0$ it remains stable [unstable] for all $T\ge 0\xb7$

(b) Suppose that $F\left(y\right)=0$ has at least one positive root and each positive root is simple. As T increases, stability switches may occur. There exists a positive ${T}^{*}$ such that (*) is unstable for all $T>{T}^{*}$. As T varies from 0 to ${T}^{*}$, at most a finite number of stability switches may occur.

##### MSC:

34K20 | Stability theory of functional-differential equations |

30C15 | Zeros of polynomials, etc. (one complex variable) |