On the zeros of transcendental functions with applications to stability of delay differential equations with two delays.

*(English)*Zbl 1068.34072The authors consider a function $h(\lambda ,\mu )$ that is analytic in $\lambda \in \u2102$ and continuous in $(\lambda ,\mu )\in \u2102\times B$, where $B\subset {\mathbb{R}}^{n}$ is open and connected. They prove a theorem on the zeros of $h$ located in the right (complex) half plane. This result is applied to the characteristic equation

$$\lambda =-b[{e}^{-\lambda {\tau}_{1}}+{e}^{-\lambda {\tau}_{2}}]-a$$

characterizing the stability behavior of the linear differential delay equation

$$\frac{dx}{dz}=-ax\left(t\right)-b[x(t-{\tau}_{1})+x(t-{\tau}_{2})]\xb7$$

By this way, the authors study stability and bifurcation of a scalar equation with two delays modeling compound optical resonators.

Reviewer: Klaus R. Schneider (Berlin)