Ordinary differential equations and descriptive set theory: uniqueness and globality of solutions of Cauchy problems in one dimension

Alessandro Andretta; Alberto Marcone

Fundamenta Mathematicae (1997)

  • Volume: 153, Issue: 2, page 157-190
  • ISSN: 0016-2736

Abstract

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We study some natural sets arising in the theory of ordinary differential equations in one variable from the point of view of descriptive set theory and in particular classify them within the Borel hierarchy. We prove that the set of Cauchy problems for ordinary differential equations which have a unique solution is -complete and that the set of Cauchy problems which locally have a unique solution is -complete. We prove that the set of Cauchy problems which have a global solution is -complete and that the set of ordinary differential equations which have a global solution for every initial condition is -complete. We prove that the set of Cauchy problems for which both uniqueness and globality hold is -complete.

How to cite

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Andretta, Alessandro, and Marcone, Alberto. "Ordinary differential equations and descriptive set theory: uniqueness and globality of solutions of Cauchy problems in one dimension." Fundamenta Mathematicae 153.2 (1997): 157-190. <http://eudml.org/doc/212220>.

@article{Andretta1997,
abstract = {We study some natural sets arising in the theory of ordinary differential equations in one variable from the point of view of descriptive set theory and in particular classify them within the Borel hierarchy. We prove that the set of Cauchy problems for ordinary differential equations which have a unique solution is $∏^0_2$-complete and that the set of Cauchy problems which locally have a unique solution is $∑^0_3$-complete. We prove that the set of Cauchy problems which have a global solution is $∑_0^4$-complete and that the set of ordinary differential equations which have a global solution for every initial condition is $∏^0_3$-complete. We prove that the set of Cauchy problems for which both uniqueness and globality hold is $∏^0_2$-complete.},
author = {Andretta, Alessandro, Marcone, Alberto},
journal = {Fundamenta Mathematicae},
keywords = {classification of sets of Cauchy problems; Borel hierarchy},
language = {eng},
number = {2},
pages = {157-190},
title = {Ordinary differential equations and descriptive set theory: uniqueness and globality of solutions of Cauchy problems in one dimension},
url = {http://eudml.org/doc/212220},
volume = {153},
year = {1997},
}

TY - JOUR
AU - Andretta, Alessandro
AU - Marcone, Alberto
TI - Ordinary differential equations and descriptive set theory: uniqueness and globality of solutions of Cauchy problems in one dimension
JO - Fundamenta Mathematicae
PY - 1997
VL - 153
IS - 2
SP - 157
EP - 190
AB - We study some natural sets arising in the theory of ordinary differential equations in one variable from the point of view of descriptive set theory and in particular classify them within the Borel hierarchy. We prove that the set of Cauchy problems for ordinary differential equations which have a unique solution is $∏^0_2$-complete and that the set of Cauchy problems which locally have a unique solution is $∑^0_3$-complete. We prove that the set of Cauchy problems which have a global solution is $∑_0^4$-complete and that the set of ordinary differential equations which have a global solution for every initial condition is $∏^0_3$-complete. We prove that the set of Cauchy problems for which both uniqueness and globality hold is $∏^0_2$-complete.
LA - eng
KW - classification of sets of Cauchy problems; Borel hierarchy
UR - http://eudml.org/doc/212220
ER -

References

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  1. [1] H. Becker, Descriptive set theoretic phenomena in analysis and topology, in: Set Theory of the Continuum, H. Judah, W. Just and H. Woodin (eds.), Math. Sci. Res. Inst. Publ. 26, Springer, 1992, 1-25. Zbl0786.04001
  2. [2] G. Beer, Topologies on Closed and Closed Convex Sets, Kluwer, 1993. 
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  4. [4] A. Kanamori, The emergence of descriptive set theory, in: Essays on the Development of the Foundations of Mathematics, J. Hintikka (ed.), Kluwer, 1995, 241-262. 
  5. [5] A. S. Kechris, Classical Descriptive Set Theory, Springer, 1995. 
  6. [6] A. Lasota and J. A. Yorke, The generic property of existence of solutions of differential equations in Banach space, J. Differential Equations 13 (1973), 1-12. Zbl0259.34070
  7. [7] Y. N. Moschovakis, Descriptive Set Theory, North-Holland, 1980. 
  8. [8] W. Orlicz, Zur Theorie der Differentialgleichung y' = f(x,y), Bull. Internat. Acad. Polon. Sci. Lettres Sér. A Sci. Math. 1932, 221-228. Zbl0006.30401
  9. [9] S. G. Simpson, Which set existence axioms are needed to prove the Cauchy/Peano theorem for ordinary differential equations?, J. Symbolic Logic 49 (1984), 783-802. Zbl0584.03039

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