The ordinary differential equation defined by a computable function whose maximal interval of existence is non-computable

Graça, Daniel; Zhong, Ning; Buescu, Jorge

http://hdl.handle.net/10400.1/1006

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Authors

Graça, Daniel

Zhong, Ning

Buescu, Jorge

Abstract(s)

Let (®, ¯) ½ R denote the maximal interval of existence of solution for the initial-value problem ½ dx dt = f(t, x), f : E ! Rm,E is an open subset of Rm+1 x(t0) = x0, with (t0, x0) 2 E. We show that (®, ¯) is r.e. (recursively enumerable) open and the solution x(t) defined on (®, ¯) is computable, provided that (a) f is computable and effectively locally Lipschitz, and (b) (t0, x0) is a computable point. We also prove that this result is the best in the sense that, for some initial-value problems satisfying (a) and (b), their maximal intervals of existence are non-recursive.