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].
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
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).
Amann, H.: On the number of solutions of asymptotically superlinear two point boundary value problems. Arch.Rational Mech. Anal. 55 (1974), 207–213.
Amann, H.: Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces. SIAM Review 18 (1976), 620–709.
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.
Dancer, E.N.: Global solution branches for positive maps. Arch.Rational Mech. Anal. 52 (1973), 181–192.
Fiacco, A.V. and MacCormick, G.P.: Nonlinear programming: sequential unconstraint minimization techniques. New York, John Wiley 1968.
Georg, K.: On the convergence of an inverse iteration method for nonlinear elliptic eigenvalue problems. To appear in Numer. Math. (1979).
Haselgrove, C.: Solution of nonlinear equations and of differential equations with two-point boundary conditions. Comput. J. 4 (1961), 255–259.
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.
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.
Lackner, K.: Computation of ideal MHD equilibria. Computer Physics Communications 12 (1976), 33–44.
Laetsch, T.: The number of solutions of a nonlinear two point boundary value problem. Indiana Univ. J. Math. 20 (1970), 1–13.
Menzel, R. and Schwetlick, H.: Zur Lösung parameterabhängiger nichtlinearer Gleichungen mit singulären Jacobi-Matrizen. Numer. Math. 30 (1978), 65–79.
Meyer-Spasche, R.: Numerical treatment of Dirichlet problems with several solutions. ISNM 31, Basel and Stuttgart, Birkhäuser 1976.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Copyright information
© 1979 Springer Basel AG
About this chapter
Cite this chapter
Georg, K. (1979). An Iteration Method for Solving Nonlinear Eigenvalue Problems. In: Albrecht, J., Collatz, L., Kirchgässner, K. (eds) Constructive Methods for Nonlinear Boundary Value Problems and Nonlinear Oscillations. International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Serie Internationale D’Analyse Numerique, vol 48. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-6283-7_3
Download citation
DOI: https://doi.org/10.1007/978-3-0348-6283-7_3
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-7643-1098-1
Online ISBN: 978-3-0348-6283-7
eBook Packages: Springer Book Archive