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Abstract

In a recent paper [7] the convergence of an inverse iteration type algorithm for a certain class of nonlinear elliptic eigenvalue problems was discussed. Such algorithms have been used successfully in plasma physics [11], but no satisfactory theoretical justification of convergence was known. While in [7] only the nondiscretized case was discussed, here an analogous algorithm for nonlinear eigenvalue problems in ℝN will be treated. This algorithm is interesting in itself, but can also be interpreted as a suitably discretized version of the algorithm discussed in [7].

Keywords

Eigenvalue Problem Spectral Radius Minimal Solution Nonlinear Programming Problem 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.

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References

  1. 1.
    Allgower, E. and Georg, K.: Simplicial and continuation methods for approximating fixed points and solutions to systems of equations. To appear in SIAM Review (1979).Google Scholar
  2. 2.
    Amann, H.: On the number of solutions of asymptotically superlinear two point boundary value problems. Arch.Rational Mech. Anal. 55 (1974), 207–213.CrossRefGoogle Scholar
  3. 3.
    Amann, H.: Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces. SIAM Review 18 (1976), 620–709.CrossRefGoogle Scholar
  4. 4.
    Crandall, M.G. and Rabinowitz, P.H.: Some continuation and variational methods for positive solutions of nonlinear elliptic eigenvalue problems. Arch. Rational Mech. Anal. 58 (1975), 207–218.CrossRefGoogle Scholar
  5. 5.
    Dancer, E.N.: Global solution branches for positive maps. Arch.Rational Mech. Anal. 52 (1973), 181–192.CrossRefGoogle Scholar
  6. 6.
    Fiacco, A.V. and MacCormick, G.P.: Nonlinear programming: sequential unconstraint minimization techniques. New York, John Wiley 1968.Google Scholar
  7. 7.
    Georg, K.: On the convergence of an inverse iteration method for nonlinear elliptic eigenvalue problems. To appear in Numer. Math. (1979).Google Scholar
  8. 8.
    Haselgrove, C.: Solution of nonlinear equations and of differential equations with two-point boundary conditions. Comput. J. 4 (1961), 255–259.CrossRefGoogle Scholar
  9. 9.
    Jeppson, M.M.: A search for the fixed points of a continuous mapping. In: Mathematical topics in economics theory and computation, R.H. Day and S.M. Robinson (eds.), 1972, 122-129.Google Scholar
  10. 10.
    Keller, H.B.: Numerical solution of bifurcation and nonlinear eigenvalue problems. In: Applications of bifurcation theory, P.H. Rabinowitz (ed.), New York, Academic Press 1977, 359–384.Google Scholar
  11. 11.
    Lackner, K.: Computation of ideal MHD equilibria. Computer Physics Communications 12 (1976), 33–44.CrossRefGoogle Scholar
  12. 12.
    Laetsch, T.: The number of solutions of a nonlinear two point boundary value problem. Indiana Univ. J. Math. 20 (1970), 1–13.CrossRefGoogle Scholar
  13. 13.
    Menzel, R. and Schwetlick, H.: Zur Lösung parameterabhängiger nichtlinearer Gleichungen mit singulären Jacobi-Matrizen. Numer. Math. 30 (1978), 65–79.CrossRefGoogle Scholar
  14. 14.
    Meyer-Spasche, R.: Numerical treatment of Dirichlet problems with several solutions. ISNM 31, Basel and Stuttgart, Birkhäuser 1976.Google Scholar

Copyright information

© Springer Basel AG 1979

Authors and Affiliations

  • Kurt Georg
    • 1
  1. 1.Institut für Angewandte MathematikUniversität BonnBonnGermany

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