Encyclopedia of Complexity and Systems Science

2009 Edition
| Editors: Robert A. Meyers (Editor-in-Chief)

Existence and Uniqueness of Solutions of Initial Value Problems

  • Gianne Derks
Reference work entry
DOI: https://doi.org/10.1007/978-0-387-30440-3_193

Definition of the Subject

Many problems in physics, engineering, biology, economics, etc., can be modeled as relationsbetween observables or states and their derivatives, hence as differential equations. When onlyderivatives with respect to one variable play a role, the differential equation is called anordinary differential equation. The field of differential equations has a long history, startingwith Newton and Leibniz in the seventeenth century. In the beginning of the study of differentialequations, the focus is on finding explicit solutions as the emphasis is on solving the underlyingphysical problems. But soon one starts to wonder: If a starting point for a solution ofa differential equation is given, does the solution always exist? And if such a solution exists, howlong does it exist and is there only one such solution? These are the questions of existence anduniqueness of solutions of initial value problems. The first existence result is given in the middleof the nineteenth...

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Primary Literature

  1. 1.
    Bellman R (1943) The stability of solutions of linear differential equations. Duke MathJ 10:643–647MathSciNetzbMATHGoogle Scholar
  2. 2.
    Carathéodory C (1918) Vorlesungen über Reelle Funktionen. Teubner, Leipzig (reprinted:(1948) Chelsea Publishing Company, New York)Google Scholar
  3. 3.
    Cauchy AL (1888) Oevres complètes (1) 6. Gauthiers-Villars, ParisGoogle Scholar
  4. 4.
    Chow S-N, Hale JK (1982) Methods of bifurcation theory. Springer, New YorkzbMATHGoogle Scholar
  5. 5.
    Coddington EA, Levinson N (1955) Theory of Ordinary Differential Equations. McGraw-Hill,New YorkzbMATHGoogle Scholar
  6. 6.
    Filippov AF (1988) Differential equations with discontinuous righthand sides. Kluwer,DordrechtGoogle Scholar
  7. 7.
    Flugge-Lotz I (1953) Discontinuous automatic control. Princeton University Press, PrincetonGoogle Scholar
  8. 8.
    Golubitsky M, Stewart I, Schaeffer DG (1985–1988) Singularities and groups inbifurcation theory, vol 1 and 2. Springer, New YorkGoogle Scholar
  9. 9.
    Gronwall TH (1919) Note on the derivative with respect to a parameter of the solutionsof a system of differential equations. Ann Math 20:292–296MathSciNetzbMATHGoogle Scholar
  10. 10.
    Guckenheimer J, Holmes P (1983) Nonlinear oscillations, dynamical systems, andbifurcations of vector fields. Springer, New YorkzbMATHGoogle Scholar
  11. 11.
    Hale JK (1969) Ordinary Differential Equations. Wiley, New YorkzbMATHGoogle Scholar
  12. 12.
    Hale JK, Verduyn Lunel SM (1993) Introduction to Functional Differential Equations.Springer, New YorkzbMATHGoogle Scholar
  13. 13.
    Hartman P (1973) Ordinary Differential Equations. Wiley, BaltimoreGoogle Scholar
  14. 14.
    Iooss G, Joseph DD (1980) Elementary stability and bifurcation theory. Springer, NewYorkzbMATHGoogle Scholar
  15. 15.
    Kneser H (1923) Ueber die Lösungen eines Systems gewöhnlicher Differentialgleichungendas der Lipschitzschen Bedingung nicht genügt. S-B Preuss Akad Wiss Phys-Math Kl 171–174Google Scholar
  16. 16.
    Kuznetsov YA (1995) Elements of applied bifurcation analysis. Springer, New YorkGoogle Scholar
  17. 17.
    Lee B, Markus L (1967) Optimal Control Theory. Wiley, New YorkzbMATHGoogle Scholar
  18. 18.
    Lindelöf ME (1894) Sur l'application de la méthode des approximations successives auxéquations différentielles ordinaires du premier ordre. Comptes rendushebdomadaires des séances de l'Académie des sciences 114:454–457Google Scholar
  19. 19.
    Müller M (1928) Beweis eines Satzes des Herrn H. Kneser über die Gesamtheit derLösungen, die ein System gewöhnlicher Differentialgleichungen durch einen Punkt schickt. Math Zeit28:349–355Google Scholar
  20. 20.
    Peano G (1890) Démonstration de l'integrabilité des équations differentiellesordinaires. Math Ann 37:182–228MathSciNetzbMATHGoogle Scholar
  21. 21.
    Picard É (1890) Mémoire sur la théorie de équations aux dérivées partielles et laméthode des approximations successives. J Math, ser 4, 6:145–210Google Scholar
  22. 22.
    Rauch J (1991) Partial Differential Equations. Springer, New YorkzbMATHGoogle Scholar
  23. 23.
    Rudin W (1976) Principles of Mathematical Analysis, 3rd edn. McGraw-Hill, New YorkzbMATHGoogle Scholar
  24. 24.
    Rudin W (1987) Real and Complex Analysis, 3rd edn. McGraw-Hill, New YorkzbMATHGoogle Scholar
  25. 25.
    Zeidler E (1995) Applied functional analysis. Vol 1 Applications to mathematicalphysics. Springer, New YorkGoogle Scholar
  26. 26.
    Clay Mathematics Institute (2000) The Millenium Problems: Navier Stokes equation.http://www.claymath.org/millennium/Navier-Stokes_Equations/

Books and Reviews

  1. 27.
    Arnold VI (1992) Ordinary Differential Equations. Springer, BerlinGoogle Scholar
  2. 28.
    Arrowsmith DK, Place CM (1990) An Introduction to Dynamical Systems. CambridgeUniversity Press, CambridgezbMATHGoogle Scholar
  3. 29.
    Braun M (1993) Differential Equations and their Applications. Springer, BerlinzbMATHGoogle Scholar
  4. 30.
    Brock WA, Malliaris AG (1989) Differential Equations, stability and chaos in DynamicEconomics. Elsevier, AmsterdamzbMATHGoogle Scholar
  5. 31.
    Grimshaw R (1993) Nonlinear Ordinary Differential Equations. CRC Press, Boca RatonGoogle Scholar
  6. 32.
    Ince EL (1927) Ordinary differential equations. Longman, Green, New YorkzbMATHGoogle Scholar
  7. 33.
    Jordan DW, Smith P (1987) Nonlinear Differential Equations. Oxford University Press,OxfordzbMATHGoogle Scholar
  8. 34.
    Werner H, Arndt H (1986) Gewöhnliche Differentialgleichungen: eine Einführung inTheorie und Praxis. Springer, BerlinGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  • Gianne Derks
    • 1
  1. 1.Department of MathematicsUniversity of SurreyGuildfordUK