# 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

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topAndretta, 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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- [5] A. S. Kechris, Classical Descriptive Set Theory, Springer, 1995.
- [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] Y. N. Moschovakis, Descriptive Set Theory, North-Holland, 1980.
- [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] 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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