  This paper introduces a natural extension of Kolchin's differential Galois
theory to positive characteristic iterative differential fields, generalizing
to the non-linear case the iterative Picard-Vessiot theory recently developed
by Matzat and van der Put. We use the methods and framework provided by the
model theory of iterative differential fields. We offer a definition of
strongly normal extension of iterative differential fields, and then prove that
these extensions have good Galois theory and that a G-primitive element theorem
holds. In addition, making use of the basic theory of arc spaces of algebraic
groups, we define iterative logarithmic equations, finally proving that our
strongly normal extensions are Galois extensions for these equations.
