Bruce van Brunt, Jonathan C. Marshall
Abstract:
In this paper we study holomorphic solutions to linear first-order
functional differential equations that have a nonlinear functional
argument. We focus on the existence of local solutions at a fixed
point of the functional argument and the holomorphic continuation of
these solutions. We show that the Julia set for the functional
argument dominates not only the conditions for holomorphic
continuation, but also the existence of local solutions.
In particular, nonconstant holomorphic solutions in a neighbourhood
of a repelling or neutral fixed point are uncommon in that the
functional argument must satisfy conditions that force it to have an
exceptional point in the former case, and a Siegel fixed point in the
latter case. In contrast, local holomorphic solutions always exist
near attracting fixed points. In this case a subset of the Julia set
forms a natural boundary for holomorphic continuation.
Published April 20, 2005.
Math Subject Classifications: 30D05, 34M05, 37F10, 37F50
Key Words: Complex functional differential equations;
pantograph equation.
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| Bruce van Brunt Department of Mathematics Massey University, New Zealand email: B.vanBrunt@massey.ac.nz |
| Jonathan C. Marshall Department of Mathematics Massey University, New Zealand |
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