Variational methods are a familiar and well developed technique in the theory of one parameter eigenvalue problems. The main aim of this paper is to show how these various concepts can be extended and generalised to deal with multiparametric eigenvalue problems. In order to do this it is first necessary to define the spectrum, in particular the eigenvalue spectrum of a multiparametric problem. This we find we can do very conveniently by introducing the notion of the state of a multiparametric problem.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Atkinson, F.V., Multipararaeter eigenvalue problems, Vol.1, Academic Press 1972.Google Scholar
  2. [2]
    Collatz, L., Multiparametric eigenvalue problems in inner product spaces, J. Comput. System Sci. 2 (1968) 333–341.CrossRefGoogle Scholar
  3. [3]
    Dash, A.T., Joint spectra. Studia Math: XLV (1973) 225-237.Google Scholar
  4. [4]
    Goldberg, S., Unbounded linear operators, McGraw-Hill, New York 1966.Google Scholar
  5. [5]
    Roach, G. F., Representation theorems for multiparametric problems in Hubert space, Meth. Verf. Math. Phys. 16 (1975) 157–174.Google Scholar
  6. [6]
    Roach, G. F., A Fredholm theory for multiparemetric problems, Nieuw Arch. v. Wiskunde (3), XXIV (1976) 49–76.Google Scholar
  7. [7]
    Roach, G. F. & Sleeman, B.D., Generalised multiparameter spectral theory. Function theor. meth. in partial diff. equ. Lect. Notes in Maths No. 561 (1976) 394-411, Springer.Google Scholar
  8. [8]
    Sleeman, B.D., Multiparemeter eigenvalue problems for ordinary differential equations, Bull. Inst. Poli. Jassy. 17(21) (1971) 51–60.Google Scholar
  9. [9]
    Taylor, A.E., Introduction to Functional Analysis, Wiley, New York 1958.Google Scholar

Copyright information

© Springer Basel AG 1977

Authors and Affiliations

  • G. F. Roach
    • 1
  1. 1.Department of MathematicsUniversity of StrathclydeGlasgowUK

Personalised recommendations