Hostname: page-component-77c89778f8-vpsfw Total loading time: 0 Render date: 2024-07-17T12:13:24.889Z Has data issue: false hasContentIssue false
  • English
  • Français

Completeness properties in L2 of the eigenfunctions of two semi-linear differential operators

Published online by Cambridge University Press:  24 October 2008

L. E. Fraenkel
L. E. Fraenkel
Affiliation:
University of Sussex
Get access Rights & Permissions [Opens in a new window]

Extract

This paper concerns the boundary-value problems

in which λ is a real parameter, u is to be a real-valued function in C2[0, 1], and problem (I) is that with the minus sign. (The differential operators are called semi-linear because the non-linearity is only in undifferentiated terms.) If we linearize the equations (for ‘ small’ solutions u) by neglecting , there result the eigenvalues λ = n2π2 (with n = 1,2,…) and corresponding normalized eigenfunctions

and it is well known ((2), p. 186) that the sequence {en} is complete in that it is an orthonormal basis for the real Hilbert space L2(0, 1). We shall be concerned with possible extensions of this result to the non-linear problems (I) and (II), for which non-trivial solutions (λ, u) bifurcate from the trivial solution (λ, 0) at the points {n2π2,0) in the product space × L2(0, 1). (Here denotes the real line.)

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 88 , Issue 3 , November 1980 , pp. 451 - 468
Copyright
Copyright © Cambridge Philosophical Society 1980

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Brown, K. J.A completeness theorem for a nonlinear problem. Proc. Edinburgh Math. Soc. (II) 19 (1974), 169–172.CrossRefGoogle Scholar
(2)Coddington, E. A. and Levinson, N.Theory of ordinary differential equations (New York, McGraw-Hill, 1955).Google Scholar
(3)Folland, G. B.Introduction to partial differential equations (Princeton, University Press, 1976).CrossRefGoogle Scholar
(4)Fraenkel, L. E.A numerical sequence and a family of polynomials arising from a question of completeness. Math. Proc. Cambridge Philos. Soc. 88 (1980), 469–481.CrossRefGoogle Scholar
(5)Kato, T.Perturbation theory for linear operators (Berlin, Springer, 1966).Google Scholar
(6)Milne-Thomson, L. M.Jacobian elliptic function tables (New York, Dover, 1950).Google Scholar
(7)Neöas, J.Les méthodes directes en théorie des equations elliptiques (Paris, Masson, 1967).Google Scholar
(8)Pimbley, G. H.Eigenfunction branches of nonlinear operators, and their bifurcations (Berlin, Springer, 1969).CrossRefGoogle Scholar
(9)Rabinowitz, P. H.Some global results for nonlinear eigenvalue problems. J. Functional Anal. 7 (1971), 487–513.CrossRefGoogle Scholar
(10)Rabinowitz, P. H. A survey of bifurcation theory. Article in Dynamical systems, vol. 1 (New York, Academic Press, 1976).Google Scholar
(11)Whittaker, E. T. and Watson, G. N.Modern analysis (Cambridge University Press, 1927).Google Scholar