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An Effective Cauchy-Peano Existence Theorem for Unique Solutions

It is shown in this paper that the solution of the initial value problem for a system of ordinary differential equations is computable if the following assumptions are satisfied: The time interval considered is computable, the system is continuous and computable, the initial values are computable, the system is effectively bounded, and the solution is unique. It should be mentioned that for a single ODE this follows immediately from the standard proof of Osgood’s existence theorem, but this approach is not available for systems of ODEs. The key assumption here is uniqueness of solution: a result of Pour-El’s and Richards’ shows that nonunique solutions may be noncomputable, even for a single ODE.