** Authors: **

J.P. Lessard and J.D. Mireles James

** Abstract:** A new approach to validated numerical integration
for delay differential equations is given. The method provides
mathematically rigorous existence results as well as error bounds
for both the solution and the Frechet derivative of the solution
with respect to a given past history segment. Chebyshev series are
used to discretize the problem, which is approx- imately solved using
a standard numerical scheme corrected via Newton's method. The
existence/error analysis uses a Newton-Kantorovich argument. We
present examples of the rigorous time stepping procedure, and
illustrate the use of the method in computer-assisted proofs of
existence for periodic solutions of the Mackey-Glass equation.

(submitted).