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On zeroes of some transcendental equations. (English) Zbl 0603.34069
The equations treated in this paper are of the form (*) $$P(z)+Q(z)e^{- Tz}=0$$ and arise from the consideration of delay-differential equations with a single delay T, $$T\geq 0$$. 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) $$P(0)+Q(0)\neq 0;$$ (iv) there are at most a finite number of zeroes of P(z) $$+Q(z)$$ in the right half-plane; (v) $$F(y)\equiv | P(iy)|^ 2-| Q(iy)|^ 2$$ 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 $$T\geq 0.$$
(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 $$T^*$$ 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, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)