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On zeroes of some transcendental equations. (English) Zbl 0603.34069
The equations treated in this paper are of the form (*) $P(z)+Q(z)e\sp{- Tz}=0$ 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 $Re(z)<0$ and unstable if at least one zero lies in $Re(z)>0$. 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 $Re(z)>\delta$ $(\delta >0)$ 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) $P(0)+Q(0)\ne 0;$ (iv) there are at most a finite number of zeroes of P(z) $+Q(z)$ in the right half-plane; (v) $F(y)\equiv \vert P(iy)\vert\sp 2-\vert Q(iy)\vert\sp 2$ for real y, has at most a finite number of real zeroes. Under these conditions, the following statements are true: (a) If the equation $F(y)=0$ has no positive roots, then if (*) is stable [unstable] at $T=0$ it remains stable [unstable] for all $T\ge 0.$ (b) Suppose that $F(y)=0$ has at least one positive root and each positive root is simple. As T increases, stability switches may occur. There exists a positive $T\sp*$ such that (*) is unstable for all $T>T\sp*$. As T varies from 0 to $T\sp*$, at most a finite number of stability switches may occur.

34K20Stability theory of functional-differential equations
30C15Zeros of polynomials, etc. (one complex variable)