Positive and spurious solutions of nonlinear eigenvalue problems

  • H. O. Peitgen
  • K. Schmitt
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 878)


Phase Portrait Elliptic Partial Differential Equation Solution Branch Finite Difference Approximation Nonlinear Eigenvalue Problem 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [A1]
    E. Allgower, On a discretization of y″+λyk=0, Proc. Conf. Roy. Irish Acad., New York-London, 1975Google Scholar
  2. [AH]
    A. Ambrosetti and P. Hess, Positive solutions of asymptotically linear elliptic eigenvalue problems, J. Math. Anal. Appl. 73 (1980), 411–422MathSciNetCrossRefzbMATHGoogle Scholar
  3. [An]
    N. Angelstorf, Spezielle periodische Lösungen einiger autonomer zeitverzögerter Differentialgleichungen mit Symmetrien, Dissertation, Universität Bremen, 1980Google Scholar
  4. [BD]
    W.-J. Beyn and E.J. Doedel, Stability and multiplicity of solutions to discretizations of nonlinear ordinary differential equations, (to appear in SIAM J. Sci. Stat. Comp.)Google Scholar
  5. [B]
    E. Bohl, On the bifurcation diagram of discrete analogues for ordinary bifurcation problems, Math. Meth. Appl. Sci. 1 (1979), 566–571MathSciNetCrossRefzbMATHGoogle Scholar
  6. [BB]
    K.J. Brown and H. Budin, On the existence of positive solutions for a class of semilinear elliptic boundary value problems, SIAM J. Math. Anal. 10 (1979), 875–883MathSciNetCrossRefzbMATHGoogle Scholar
  7. [C]
    L. Collatz, The numerical treatment of differential equations, Springer Verlag, Berlin 1960CrossRefzbMATHGoogle Scholar
  8. [CR]
    M. Crandall and P. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal. 8 (1971), 321–340MathSciNetCrossRefzbMATHGoogle Scholar
  9. [G]
    R. Gaines, Difference equations associated with boundary value problems for second order nonlinear ordinary differential equations, SIAM J. Numer. Anal. 11 (1974), 411–434MathSciNetCrossRefzbMATHGoogle Scholar
  10. [GT]
    D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Heidelberg-New York, 1977CrossRefzbMATHGoogle Scholar
  11. [JPS]
    H. Jürgens, H.O. Peitgen and D. Saupe, Topological perturbations in the numerical study of nonlinear eigenvalue and bifurcation problems, Proc. Conf. Analysis and Computation of Fixed Points, Academic Press, New York-London, 1980, 139–181CrossRefGoogle Scholar
  12. [IK]
    E. Isaacson and H. Keller, Analysis of Numerical Methods, John Wiley, New York, 1966zbMATHGoogle Scholar
  13. [M]
    P. Morse, Boundary Value Problems For Nonlinear Difference Equations, Ph. D. Thesis, University of Utah, Salt Lake City, 1980Google Scholar
  14. [OM]
    R.E. O'Malley, Jr., Phaseplane solutions to some singular perturbation problems, J. Math. Anal. Appl. 54 (1976), 449–466MathSciNetCrossRefzbMATHGoogle Scholar
  15. [P]
    S.I. Pohozaev, Eigenfunctions of the equation Δu + λf(u) = 0, Soviet Math. Dokl., 6 (1965), 1408–1411MathSciNetGoogle Scholar
  16. [PP]
    H.O. Peitgen and M. Prüfer, The Leray Schauder continuation method is a constructive element in the numerical study of nonlinear eigenvalue and bifurcation problems, Proc. Conf. Functional Differential Equations and Approximation of Fixed Points, Springer Lecture Notes in Math. 730 (1980), 326–409Google Scholar
  17. [PSS]
    H.O. Peitgen, D. Saupe and K. Schmitt, Nonlinear elliptic boundary value problems versus their finite difference approximations: numerically irrelevant solutions, J. reine angew. Mathematik 322 (1981), 74–117MathSciNetzbMATHGoogle Scholar
  18. [PS1]
    H.O. Peitgen and K. Schmitt, Perturbations topologiques globales des problèmes non linéaires aux valeurs propres, C.R. Acad. Sc. Paris 291 (1980), 271–274MathSciNetzbMATHGoogle Scholar
  19. [PS2]
    H.O. Peitgen and K. Schmitt, Global topological perturbations of nonlinear elliptic eigenvalue problems, (to appear) Report Nr. 33, Forschungsschwerpunkt "Dynamische Systeme", Universität Bremen 1981Google Scholar
  20. [R]
    P. Rabinowitz, Some aspects of nonlinear eigenvalue problems, Rocky Mountain J. Math. 3 (1973), 162–202MathSciNetCrossRefGoogle Scholar
  21. [Sa]
    D. Saupe, On accelerating PL continuation algorithms by predictor corrector methods, (to appear in Math.Prog., Report Nr. 22, Forschungsschwerpunkt "Dynamische Systeme", Universität Bremen, 1981)Google Scholar
  22. [Sc]
    J. Schröder, M-matrices and generalizations using an operator theory approach, SIAM Review 20 (1978), 213–244MathSciNetCrossRefzbMATHGoogle Scholar
  23. [SA]
    H. Spreuer and E. Adams, Pathalogische Beispiele von Differenzenverfahren bei nichtlinearen gewöhnlichen Randwertaufgaben, ZAMM 57 (1977), T 304–T 305MathSciNetzbMATHGoogle Scholar
  24. [SS]
    A.B. Stephens and G.R. Shubin, Multiple solutions and bifurcation of finite difference approximations to some steady state problems of fluid dynamics, preprint of Naval Surface Weapons Center, 1981Google Scholar
  25. [St]
    C.A. Stuart, Concave solutions of singular non-linear differential equations, Math. Z. 136 (1974), 117–135MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 1981

Authors and Affiliations

  • H. O. Peitgen
    • 1
  • K. Schmitt
    • 2
  1. 1.Forschungsschwerpunkt "Dynamische Systeme" Fachbereich MathematikUniversität BremenBremen 33
  2. 2.Department of MathematicsUniversity of UtahSalt Lake CityUSA

Personalised recommendations