Unique summing of formal power series solutions to advanced and delayed differential equations
David W. Pravica - 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.
Received: September 2004; Revised: May 2005; Published: September 2005. |