Holomorphic solutions to functional differential equations
This paper is concerned with the initial value problem \[ y'(z)+\alpha y(z)=\lambda y(z^n), \qquad y(\widetilde{z})=y_0, \] where \(\alpha\), \(\lambda\neq 0\) and \(y_0\) are constants, \(n>1\) is an integer and \(\widetilde{z}\) is a fixed point for \(g(z)=z^n\). The function \(g\) has one attracting fixed point at \(z_0=0\) and \(n-1\) repelling fixed points on the circle \(C(0,1)=\{z:|z|=1\}\). First, the authors study solutions that are holomorphic at the fixed points (attracting or repelling). More especially, they prove the existence of a natural boundary for solutions holomorphic at the attracting fixed point and also they show that the holomorphic solutions at repelling fixed points can be continued throughout the complex plane except at attracting fixed points. The special case \(\alpha=0\) is studied separately. This work is motivated by the paper of \textit{M. Heard} [J. Math. Anal. Appl. 44, 745--757 (1973; Zbl 0289.34115)] which deals with the famous pantograph equation.
complex functional differential equations
scientific article; zbMATH DE number 5710091