A class of analytic advanced and delayed differential equations, which
are defined in a neighborhood of an initial point, and which are assumed to have
formal solutions in terms of power series, is studied. We provide growth conditions
whereby the (perhaps non-convergent) formal series solutions can be extended to
analytic solutions defined on a sectorial domain with vertex at the initial point.
By introducing a new Laplace-Borel kernel, and obtaining estimates on its decay
rate, the concept of a Gevrey series is generalized. The class of equations studied
includes advanced and delayed initial value problems with polynomial coefficients.
Key estimates are shown and an example of a new application is given.
Mathematics Subject Classification: Primary: 34M25, 34M30; Secondary: 40G10, 44A10.