An ordinary differential equation is a differential equation involving derivatives of a function with respect to one argument only, i.e. the function is on a manifold only of dimension . This function can be vector valued, what is sometimes viewed as a system of possibly coupled equations; still all of them have the derivatives taken with respect to the same parameter.
A basic theorem concerns existence and uniqueness of local solutions to initial value problems. Let be a Banach space; denote the ball of radius about by . Given and , put .
(Picard-Lindelöf) Suppose is a continuous function satisfying the following conditions:
(Lipschitz condition) There is a Lipschitz constant such that
for all ;
(Boundedness) There is a constant such that .
Then for any and , there exists exactly one solution to the initial value problem