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## Fundamenta Mathematicae

1997 | 153 | 2 | 157-190
Tytuł artykułu

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

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Słowa kluczowe
Czasopismo
Rocznik
Tom
Numer
Strony
157-190
Opis fizyczny
Daty
wydano
1997
otrzymano
1996-09-15
poprawiono
1997-01-15
Twórcy
autor
• Dipartimento di Matematica, Università di Torino, via Carlo Alberto 10, 10123 Torino, Italy
autor
• Dipartimento di Matematica, Università di Torino, via Carlo Alberto 10, 10123 Torino
Bibliografia
• [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.
• [2] G. Beer, Topologies on Closed and Closed Convex Sets, Kluwer, 1993.
• [3] L. Faina, Uniqueness and continuous dependence of the solutions for functional differential equations as a generic property, Nonlinear Anal. 23 (1994) 745-754.
• [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] 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.
• [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.
• [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.
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