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

Pages: 730 - 737, Issue Special, August 2005

 Abstract        Full Text (213.6K)              

David W. Pravica - Department of Mathematics, East Carolina University, Greenville, NC 27858, United States (email)
Michael J. Spurr - Department of Mathematics, East Carolina University, Greenville, NC 27858, United States (email)

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.

Received: September 2004;      Revised: May 2005;      Published: September 2005.