Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Solvability of a finite or infinite system of discontinuous quasimonotone differential equations

Authors: Daniel C. Biles and Eric Schechter
Journal: Proc. Amer. Math. Soc. 128 (2000), 3349-3360
MSC (2000): Primary 34A12, 34A40; Secondary 45G15
Published electronically: May 18, 2000
MathSciNet review: 1707137
Full-text PDF

Abstract | References | Similar Articles | Additional Information


This paper proves the existence of solutions to the initial value problem

\begin{displaymath}(\mathrm{IVP})\qquad\qquad\left\{\begin{array}{l} x'(t)=f(t,x(t))\qquad\quad (0\le t\le 1), x(0)=0,\end{array}\right.\end{displaymath}

where $f:[0,1]\times \mathbb{R} ^M\to \mathbb{R} ^M$ may be discontinuous but is assumed to satisfy conditions of superposition-measurability, quasimonotonicity, quasisemicontinuity, and integrability. The set $M$ can be arbitrarily large (finite or infinite); our theorem is new even for $\mbox{card}(M)=2$. The proof is based partly on measure-theoretic techniques used in one dimension under slightly stronger hypotheses by Rzymowski and Walachowski. Further generalizations are mentioned at the end of the paper.

References [Enhancements On Off] (What's this?)

  • 1. M. Balcerzak, Some results on superposition-measurability, Real Analysis Exchange 17 (1991/ 1992), 597-607. MR 93k:28001
  • 2. M. Balcerzak and K. Ciesielski, On the sup-measurable functions problem, Real Anal. Exchange 23 (2) (1997-98), 787-797. CMP 98:17
  • 3. Daniel C. Biles, Existence of solutions for discontinuous differential equations, Differential and Integral Equations 8 (1995), 1525-1532. MR 96d:34003
  • 4. Daniel C. Biles and Paul A. Binding, On Carathéodory's conditions for the initial value problem, Proceedings of the American Mathematical Society 125 (1997), 1371-1376. MR 97g:34008
  • 5. L. P. Burton and William M. Whyburn, Minimax solutions of ordinary differential systems, Proceedings of the American Mathematical Society 3 (1952), 794-803. MR 14:470i
  • 6. C. Carathéodory, Vorlesungen über reelle Funktionen, Leipzig, 1927; reprinted, New York, 1948, pp. 665-688. MR 37:1530
  • 7. Earl A. Coddington and Norman Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955. MR 16:1022b
  • 8. J. Dieudonné, Deux examples singuliers d'équations différentielles, Acta Sci. Math. (Szeged) 12B (1950), 38-40. MR 11:729d
  • 9. N. Dunford and J. T. Schwartz, Linear Operators, Part I: General Theory, Pure Appl. Math. 7, Wiley Interscience, New York, 1957. MR 22:8302
  • 10. G. S. Goodman, Subfunctions and the initial-value problem for differential equations satisfying Carathéodory's hypotheses, Journal of Differential Equations 7 (1970), 232-242. MR 41:540
  • 11. Seppo Heikkilä and V. Lakshmikantham, On first order differential equations in ordered Banach spaces, in Inequalities and Applications, ed. by R. P. Agarwal, World Scientific, New Jersey, 1994, p. 293-301. MR 95g:34095
  • 12. E. Kamke, Zur Theorie der Systeme gewöhnlicher Differentialgleichungen II, Acta Mathematica 58 (1932), 57-85.
  • 13. A. B. Kharazishvili, Sup-measurable and weakly sup-measurable mappings in the theory of ordinary differential equations, J. Appl. Anal. 3 (1997), 211-223. MR 99e:28003
  • 14. M. A. Krasnosel'ski{\u{\i}}\kern.15em, P. P. Zabre{\u{\i}}\kern.15emko, et al., Integral operators in spaces of summable functions (Russian), ``Nauka,'' Moscow, 1966. Translated to English by T. Ando, in Monographs and Textbooks on Mechanics of Solids and Fluids, Mechanics: Analysis. Noordhoff International Publishing, Leiden, 1976. MR 52:6505
  • 15. Roland Lemmert, Existenzsätze für gewöhnliche Differentialgleichungen in geordneten Banachräumen, Funkcialaj Ekvacioj 32 (1989), 243-249. MR 90i:34096
  • 16. Max Müller, Über das Fundamentaltheorem in der Theorie der gewöhnlichen Differentialgleichungen, Mathematische Zeitschrift 26 (1926), 619-645.
  • 17. G. Peano, Sull'inegrabilitá delle equazione differenziali di primo ordine, Atti della Accademia delle Scienze di Torino 21 (1885-1886), 677-685.
  • 18. O. Perron, Ein neuer Existenzbeweis für die Integrale der Differentialgleichung $y'=f(x,y)$, Math. Annl. 76 (1914), 471-484.
  • 19. Irene Redheffer and Peter Volkmann, Ein Fixpunktsatz für quasimonoton wachsende Funktionen, Arch. der Math. 70 (1998), 307-312. MR 99a:47086
  • 20. Ray M. Redheffer and Wolfgang Walter, Flow-invariant sets and differential inequalities in normed spaces, Applicable Analysis 5 (1975), 149-161. MR 57:10155
  • 21. Witold Rzymowski and Dariusz Walachowski, One-dimensional differential equations under weak assumptions, Journal of Mathematical Analysis and Applications 198 (1996), 657-670. MR 97c:34009
  • 22. E. Schechter, A survey of local existence theories for abstract nonlinear initial value problems, pp. 136-184 in Nonlinear Semigroups, Partial Differential Equations, and Attractors (proc. sympos. Washington 1987), ed. by T. L. Gill and W. W. Zachary, Lecture Notes in Math. 1394, Springer-Verlag, Berlin, 1989. MR 91c:34020
  • 23. E. Schechter, Handbook of Analysis and its Foundations, Academic Press, San Diego, 1997, pp. 824-826. MR 98b:00009
  • 24. Alice Chaljub-Simon, Roland Lemmert, Sabina Schmidt, and Peter Volkmann, Gewöhnliche Differentialgleichungen mit quasimonoton wachsenden rechten Seiten in geordneten Banachräumen, pp. 307-320 in General Inequalities 6, ed. by Wolfgang Walter, International Series of Numerical Mathematics 103, Birkhäuser Verlag, Basel, 1992. MR 94c:34092
  • 25. Alice Simon and Peter Volkmann, Remark on quasimonotonicity, in Inequalities and Applications, ed. by R. P. Agarwal, World Scientific, New Jersey, 1994, p. 543-548. MR 95h:34024
  • 26. J. W. Sragin, Conditions for measurability of superpositions (Russian), Dokl. Akad. Nauk SSSR 197 (1971), 295-298; translated in Soviet Math. Dokl. 12 (1971), 465-470.
  • 27. J. Szarski, Sur un système d'inéqalités différentielles, Ann. Soc. Polon. Math. 20 (1947), 126-134. MR 10:121g
  • 28. J. Szarski, Sur les systèmes majorants d'équations différentielles ordinaires, Ann. Soc. Polon. Math. 23 (1950), 206-223. MR 12:705b
  • 29. Peter Volkmann, Gewöhnliche Differentialungleichungen mit quasimonoton wachsenden Funktionen in topologischen Vektorraümen, Mathematische Zeitschrift 127 (1972), 157-164. MR 46:7661
  • 30. Wolfgang Walter, Differential and Integral Inequalities, Springer-Verlag, New York, 1970. Translation of Differential- und Integral-Ungleichungen, Springer, Berlin, 1964. MR 42:6391
  • 31. W. Walter, Ordinary Differential Equations, Graduate Texts in Mathematics 182, Springer, New York, 1998. CMP 98:15
  • 32. T. Wazewski, Systèmes des équations et des inégalités différentielles ordinaires aux deuxièmes membres monotones et leurs applications, Ann. Soc. Polon. Math. 23 (1950), 112-166. MR 12:705a

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 34A12, 34A40, 45G15

Retrieve articles in all journals with MSC (2000): 34A12, 34A40, 45G15

Additional Information

Daniel C. Biles
Affiliation: Department of Mathematics, Western Kentucky University, Bowling Green, Kentucky 42101-3576

Eric Schechter
Affiliation: Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37240-0001

Keywords: Quasimonotone, semicontinuous, quasisemicontinuous, supmeasurable, subsolution
Received by editor(s): January 13, 1999
Published electronically: May 18, 2000
Communicated by: Hal L. Smith
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society