Unique summing of formal power series solutions to advanced and delayed differential equations

David W. Pravica; Michael J. Spurr

doi:10.3934/proc.2005.2005.730

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2005, Volume 2005: 730-737. Doi: 10.3934/proc.2005.2005.730 open access

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Unique summing of formal power series solutions to advanced and delayed differential equations

David W. Pravica¹ and
Michael J. Spurr^1,

1.
Department of Mathematics, East Carolina University, Greenville, NC 27858

Received: September 2004

Revised: May 2005

Published: August 2005

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Abstract

The analytic delayed-differential equation $z^2 \psi ^{\ \! \prime } (z) \ + \ \psi (z/q) \ = \ z$ for $q>1$ has a solution which can be expressed as a formal power series. A $q$-advanced Laplace-Borel kernel provides for the construction of an analytic solution whose domain is the right half plane with vertex at the initial point $z=0$. This method is extended to provide a continuous family of solutions, of which a subfamily extends to a punctured neighborhood of $z=0$ on the logarithmic Riemann surface. Conditions are given on the asymptotics of $\psi ^{\ \! \prime } (z)$ near $z=0$ to ensure uniqueness.

Keywords:

Delay equations,
q-advanced Gevrey asymptotics.

Mathematics Subject Classification: 34M25, 34M30; 40C10, 40G10, 44A10.

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