Numerische Mathematik

, Volume 20, Issue 5, pp 356–363 | Cite as

Iterationsverfahren für monotone, nicht notwendig Lipschitz-beschränkte Operatoren im Hilbert-Raum

  • A. Sachs
Article
Received:
  • 30 Downloads

Summary

We state three theorems concerning iterative methods for solving equations containing monotone, but not necessarily uniformly monotone operators in Hilbert space, which may be not Lipschitz bounded, thus extending results obtained by Sibony [9, 10]. We apply the method to a nonlinear elliptic Dirichlet problem of laminar flow theory of non Newtonian fluids, thus improving the rather experimental treatment given in Ames [1] and removing the condition of nonzero difference quotients assumed by Cryer [3].

This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Literatur

  1. 1.
    Ames, W.: Nonlinear partial differential equations in engineering. New York: Academic Press 1965.Google Scholar
  2. 2.
    Browder, F.: The solvability of non-linear functional equations. Duke Math. J.30, 557–566 (1963).Google Scholar
  3. 3.
    Cryer, C.: On the numerical solution of a quasi-linear elliptic equation. J. ACM14, 363–375 (1967).Google Scholar
  4. 4.
    D'Yakonov, E.: On the solution of some elliptic difference equations. J. Inst. Maths Applics7, 1–20 (1971).Google Scholar
  5. 5.
    Frehse, J.: Existenz und Konvergenz von Lösungen nichtlinearer elliptischer Differenzengleichungen unter Dirichlet-Randbedingungen. Math. Z.109, 311–343 (1969).Google Scholar
  6. 6.
    Petry, W.: Ein Iterationsverfahren zum Lösen von Randwertproblemen nichtlinearer Differentialgleichungen. Computing5, 27–44 (1970).Google Scholar
  7. 7.
    Petryshyn, W.: On the extension and the solution of non linear operator equations. Ill. J. Math.10, 255–274 (1966).Google Scholar
  8. 8.
    Sachs, A.: Iterationsverfahren für elliptische (nichtlineare) Differenzenoperatoren in Divergenzform. Lecture Notes in Mathematics267, 305–322 (1972). Vorträge der Tagung “Numerische Lösung nichtlinearer partieller Differential- und Integrodifferentialgleichungen” im Mathematischen Forschungsinstitut Oberwolfach,28. 11.–4.12. 1971.Google Scholar
  9. 9.
    Sibony, M.: Méthodes itératives pour les équations et inéquations aux dérivées partielles non linéaires de type monotone. Calcolo7, 65–183 (1970).Google Scholar
  10. 10.
    Sibony, M.: Sur 1 approximation d'équations et inéquations aux dérivées partielles non linéaires de type monotone. J. Math. Anal. Appl.34, 502–564 (1971).Google Scholar

Copyright information

© Springer-Verlag 1972

Authors and Affiliations

  • A. Sachs
    • 1
  1. 1.Mathematisches InstitutUniversität MünchenMünchen 2Bundesrepublik Deutschland

Personalised recommendations