A natural boundary for solutions to the second order pantograph equation

Pantograph equations are characterized by the presence of a linear functional argument. These equations arise in several applications and often the argument has a repelling fixed point at the origin. Recently, Marshall et al. [J. Math. Anal. Appl. 268 (2002) 157–170] studied a related class of functional differential equations with nonlinear functional arguments and showed that, generically, solutions to such equations have a natural boundary. Their approach uses some well-known properties of the Julia set and relies heavily on the nonlinearity of the functional argument. The method is not directly applicable to pantograph type equations though some of the techniques can be exploited. In this paper we show that solutions to pantograph equations generally have natural boundaries. We focus on a special set of solutions that have the imaginary axis as a natural boundary.  2004 Elsevier Inc. All rights reserved