Advertisement

Differential Equations

, Volume 46, Issue 2, pp 195–207 | Cite as

On the uniqueness of the solution of nonlinear differential-algebraic systems

  • A. A. Shcheglova
Ordinary Differential Equations

Abstract

We consider the Cauchy problem for a system of nonlinear ordinary differential equations unsolved for the derivative of the unknown vector function and identically degenerate in the domain. We prove a theorem on the coincidence of two smooth solutions of the considered problem. We show that, under some additional assumptions, the above-mentioned problem cannot have classical solutions with less smoothness. We obtain conditions under which the problem has a fixed finite number of solutions.

Keywords

Cauchy Problem Implicit Function Theorem Continuous Partial Derivative Permutation Matrice Independent Column 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Petrovskii, I.G., Lektsii po teorii obyknovennykh differentsial’nykh uravnenii (Lectures on the Theory of Ordinary Differential Equations), Moscow, Izdat. Tekhn.-Teor. Lit., 1949.Google Scholar
  2. 2.
    Campbell, S.L. and Griepentrog, E., Solvability of General Differential Algebraic Equations, SIAM J. Sci. Stat. Comput., 1995, no. 16, pp. 257–270.Google Scholar
  3. 3.
    Kunkel, P. and Mehrmann, V., Regular Solutions of Nonlinear Differential-Algebraic Equations and Their Numerical Determination, Numer. Math., 1998, no. 79, pp. 581–600.Google Scholar
  4. 4.
    Chistyakov, V.F. and Shcheglova, A.A., Izbrannye glavy teorii algebro-differentsial’nykh sistem (Selected Chapters in the Theory of Differential-Algebraic Systems), Novosibirsk: Nauka, 2003.zbMATHGoogle Scholar
  5. 5.
    Shilov, G.E., Matematicheskii analiz (funktsii neskol’kikh veshchestvennykh peremennykh) (Mathematical Analysis (Functions of Several Real Variables)), Moscow, 1972, parts 1, 2.Google Scholar
  6. 6.
    Rabier, P.J. and Rheinboldt, W.C., Theoretical and Numerical Analysis of Differential-Algebraic Equations. Handbook of Numerical Analysis, vol. VIII, Amsterdam, 2002.Google Scholar
  7. 7.
    Shcheglova, A.A., Nonlinear Differential-Algebraic Systems, Sibirsk. Mat. Zh., 2007, vol. 48, no. 4, pp. 931–948.zbMATHMathSciNetGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2010

Authors and Affiliations

  • A. A. Shcheglova
    • 1
  1. 1.Institute for System Dynamics and Control Theory Siberian BranchRussian Academy of SciencesIrkutskRussia

Personalised recommendations