Editing Initial value problem (section)

== Existence and uniqueness of solutions ==

The [[Picard–Lindelöf theorem]] guarantees a unique solution on some interval containing ''t''<sub>0</sub> if ''f'' is continuous on a region containing ''t''<sub>0</sub> and ''y''<sub>0</sub> and satisfies the [[Lipschitz continuity|Lipschitz condition]] on the variable ''y''.
The proof of this theorem proceeds by reformulating the problem as an equivalent [[integral equation]]. The integral can be considered an operator which maps one function into another, such that the solution is a [[Fixed point (mathematics)|fixed point]] of the operator. The [[Banach fixed point theorem]] is then invoked to show that there exists a unique fixed point, which is the solution of the initial value problem.

An older proof of the Picard–Lindelöf theorem constructs a sequence of functions which converge to the solution of the integral equation, and thus, the solution of the initial value problem. Such a construction is sometimes called "Picard's method" or "the method of successive approximations". This version is essentially a special case of the Banach fixed point theorem.

[[Hiroshi Okamura]] obtained a [[necessary and sufficient condition]] for the solution of an initial value problem to be unique.  This condition has to do with the existence of a [[Lyapunov function]] for the system.

In some situations, the function ''f'' is not of [[Smooth function|class ''C''<sup>1</sup>]], or even [[Lipschitz continuity|Lipschitz]], so the usual result guaranteeing the local existence of a unique solution does not apply. The [[Peano existence theorem]] however proves that even for ''f'' merely continuous, solutions are guaranteed to exist locally in time; the problem is that there is no guarantee of uniqueness. The result may be found in Coddington & Levinson (1955, Theorem 1.3) or Robinson (2001, Theorem 2.6). An even more general result is the [[Carathéodory existence theorem]], which proves existence for some discontinuous functions ''f''.