Initial value problem: Difference between revisions

Revision as of 13:46, 2 February 2021

Revision as of 18:19, 6 May 2021

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For a large class of initial value problems, the existence and uniqueness of a solution can be illustrated through the use of a calculator.

The [[Picard–Lindelöf theorem]] guarantees a unique solution on some interval containing ''t''₀ if ƒ''f'' is continuous on a region containing ''t''₀ and ''y''₀ 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.

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[[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 ƒ''f'' is not of [[Smooth function|class ''C''¹]], 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 ƒ''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 ƒ''f''.

==Examples==