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.
@article{bwmeta1.element.bwnjournal-article-fmv153i2p157bwm,
author = {Alessandro Andretta and Alberto Marcone},
title = {Ordinary differential equations and descriptive set theory: uniqueness and globality of solutions of Cauchy problems in one dimension},
journal = {Fundamenta Mathematicae},
volume = {154},
year = {1997},
pages = {157-190},
zbl = {0883.04002},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-fmv153i2p157bwm}
}
Andretta, Alessandro; Marcone, Alberto. Ordinary differential equations and descriptive set theory: uniqueness and globality of solutions of Cauchy problems in one dimension. Fundamenta Mathematicae, Tome 154 (1997) pp. 157-190. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-fmv153i2p157bwm/
[00000] [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. | Zbl 0786.04001
[00001] [2] G. Beer, Topologies on Closed and Closed Convex Sets, Kluwer, 1993.
[00002] [3] L. Faina, Uniqueness and continuous dependence of the solutions for functional differential equations as a generic property, Nonlinear Anal. 23 (1994) 745-754. | Zbl 0810.34062
[00003] [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.
[00004] [5] A. S. Kechris, Classical Descriptive Set Theory, Springer, 1995.
[00005] [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. | Zbl 0259.34070
[00006] [7] Y. N. Moschovakis, Descriptive Set Theory, North-Holland, 1980.
[00007] [8] W. Orlicz, Zur Theorie der Differentialgleichung y' = f(x,y), Bull. Internat. Acad. Polon. Sci. Lettres Sér. A Sci. Math. 1932, 221-228. | Zbl 0006.30401
[00008] [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. | Zbl 0584.03039