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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Computability, noncomputability and undecidability of maximal intervals of IVPs


Authors: D.S. Graça, N. Zhong and J. Buescu
Journal: Trans. Amer. Math. Soc. 361 (2009), 2913-2927
MSC (2000): Primary 65L05, 68Q17; Secondary 03D35
Published electronically: January 22, 2009
MathSciNet review: 2485412
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Abstract: Let $ (\alpha,\beta)\subseteq \mathbb{R}$ denote the maximal interval of existence of solutions for the initial-value problem

\begin{displaymath} \left\{ \begin{array}[c]{l} \frac{dx}{dt}=f(t,x),\\ x(t_{0})=x_{0}, \end{array}\right. \end{displaymath}

where $ E$ is an open subset of $ \mathbb{R}^{m+1}$, $ f$ is continuous in $ E$ and $ (t_{0},x_{0})\in E$. We show that, under the natural definition of computability from the point of view of applications, there exist initial-value problems with computable $ f$ and $ (t_{0},x_{0})$ whose maximal interval of existence $ (\alpha,\beta)$ is noncomputable. The fact that $ f$ may be taken to be analytic shows that this is not a lack of the regularity phenomenon. Moreover, we get upper bounds for the ``degree of noncomputability'' by showing that $ (\alpha,\beta)$ is r.e. (recursively enumerable) open under very mild hypotheses. We also show that the problem of determining whether the maximal interval is bounded or unbounded is in general undecidable.


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Additional Information

D.S. Graça
Affiliation: DM/FCT, Universidade do Algarve, C. Gambelas, 8005-139 Faro, Portugal – and – SQIG, Instituto de Telecomunicações, Instituto Superior Técnico, Torre Norte, Piso 10, Av. Rovisco Pais, 1049-001 Lisboa, Portugal
Email: dgraca@ualg.pt

N. Zhong
Affiliation: Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio 45221-0025
Email: ning.zhong@uc.edu

J. Buescu
Affiliation: CAMGSD and Dep. Matemática, Faculdade de Ciências, Ed C6, Piso 2, Campo Grande, 1749-006 Lisboa, Portugal
Email: jbuescu@ptmat.fc.ul.pt

DOI: http://dx.doi.org/10.1090/S0002-9947-09-04929-0
PII: S 0002-9947(09)04929-0
Received by editor(s): July 6, 2006
Published electronically: January 22, 2009
Additional Notes: The first author thanks M. Campagnolo, O. Bournez, and E. Hainry for helpful discussions. The first author was partially supported by Fundação para a Ciência e a Tecnologia and EU FEDER POCTI/POCI via CLC, grant SFRH/BD/17436/2004, and the project ConTComp POCTI/MAT/45978/2002. Additional support was also provided by the Fundação Calouste Gulbenkian through the Programa Gulbenkian de Estímulo à Investigação.
The second author was partially supported by University of Cincinnati’s Taft Summer Research Fellowship.
The third author was partially supported by Fundação para a Ciência e a Tecnologia and EU FEDER POCTI/POCI via CAMGSD
Article copyright: © Copyright 2009 American Mathematical Society