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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, T0. 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)>δ (δ>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)0; (iv) there are at most a finite number of zeroes of P(z) +Q(z) in the right half-plane; (v) F(y)|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 T0·

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