  Fixed point theorems are one of the many tools used to prove existence and
uniqueness of differential equations. When the data involved contains products
of distributions, some of these tools may not be useful. Thus rises the
necessity to develop new environments and tools capable of handling such
situations. The foundations of a Generalized Differential Geometry is set
having Classical Differential Geometry as a discontinuous subcase, a fixed
point theorem for hypersequences is proved in the context of Colombeau
Generalized Functions and it is shown how it can be used to obtain existence
and uniqueness of differential equations whose data involve products of
distributions. Thus also setting the foundations of a Generalized Analysis. The
strain is also picked up setting the foundations of generalized manifolds and
shown that each classical manifold can be discretely embedded in a generalized
manifold in such a way that the differential structure of the latter is a
natural extension of the differential structure of the former. It is inferred
that $\ {\cal{D}}^{\prime}(\Omega)$ is discretely embedded in $\
{\cal{G}}(\Omega)$, that the elements of $\ C^{\infty}(\Omega)$ form a grid of
equidistant points in $\ {\cal{G}}(\Omega)$ and that association in $\
{\cal{G}}(\Omega)$ is a topological and not an algebraic notion. Ergo,
classical solutions to differential equations are scarce. These achievements
reckoned upon the Generalized Differential Calculus invented by the first
author and his collaborators. Hopefully, Generalized Differential Calculus and
the developments presented in this paper, may be of interest to those working
in Analysis, Applied Mathematics, Geometry and Physics.
