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PERSISTENCE OF THE NORMALIZED EIGENVECTORS OF A PERTURBED OPERATOR IN THE VARIATIONAL CASE

Published online by Cambridge University Press:  25 February 2013

RAFFAELE CHIAPPINELLI ,
MASSIMO FURI  and
MARIA PATRIZIA PERA
RAFFAELE CHIAPPINELLI
Affiliation:
Dipartimento di Scienze Matematiche ed Informatiche, Pian dei Mantellini 44, I-53100 Siena, Italy e-mail: [email protected]
MASSIMO FURI
Affiliation:
Dipartimento di Matematica Applicata ‘G. Sansone’, Via S. Marta 3, I-50139 Florence, Italy e-mails: [email protected], [email protected]
MARIA PATRIZIA PERA
Affiliation:
Dipartimento di Matematica Applicata ‘G. Sansone’, Via S. Marta 3, I-50139 Florence, Italy e-mails: [email protected], [email protected]
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Abstract

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Let H be a real Hilbert space and denote by S its unit sphere. Consider the nonlinear eigenvalue problem Ax + ε B(x) =δ x, where A: H → H is a bounded self-adjoint (linear) operator with nontrivial kernel Ker A, and B: H → H is a (possibly) nonlinear perturbation term. A unit eigenvector x0 ∈ S∩ Ker A of A (thus corresponding to the eigenvalue δ=0, which we assume to be isolated) is said to be persistent, or a bifurcation point (from the sphere S∩ Ker A), if it is close to solutions x ∈ S of the above equation for small values of the parameters δ ∈ ℝ and ε ≠ 0. In this paper, we prove that if B is a C1 gradient mapping and the eigenvalue δ=0 has finite multiplicity, then the sphere S∩ Ker A contains at least one bifurcation point, and at least two provided that a supplementary condition on the potential of B is satisfied. These results add to those already proved in the non-variational case, where the multiplicity of the eigenvalue is required to be odd.

Keywords

Primary 47A55Secondary 47J0547J1047J15
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 55 , Issue 3 , September 2013 , pp. 629 - 638
Copyright
Copyright © Glasgow Mathematical Journal Trust 2013 

References

REFERENCES

1.Berger, M. S., Nonlinearity and functional analysis (Academic Press, New York, 1977).Google Scholar
2.Chiappinelli, R., Furi, M. and Pera, M. P., Normalized eigenvectors of a perturbed linear operator via general bifurcation, Glasgow Math. J. 50 (2008), 303–318.CrossRefGoogle Scholar
3.Chiappinelli, R., Furi, M. and Pera, M. P., Topological persistence of the normalized eigenvectors of a perturbed self-adjoint operator, Appl. Math. Lett. 23 (2010), 193–197.Google Scholar
4.Chiappinelli, R., Furi, M. and Pera, M. P., A new theme in nonlinear analysis: Continuation and bifurcation of the unit eigenvectors of a perturbed linear operator, Commun. Appl. Anal. 15 (2–4) (2011), 299–312.Google Scholar
5.Crandall, M. G. and Rabinowitz, P. H., Bifurcation from simple eigenvalues, J. Funct. Anal. 8 (1971), 321–340.Google Scholar
6.Deimling, K., Nonlinear functional analysis (Springer, New York, 1985).Google Scholar
7.Furi, M., Martelli, M. and Pera, M. P., General bifurcation theory: Some local results and applications, in Differential equations and applications to biology and to industry (Cooke, K., Cumberbatch, E., Martelli, M, Tang, B. and Thieme, H., Editors) (World Scientific, River Edge, NJ, 1996), 101–115.Google Scholar
8.Furi, M. and Pera, M. P., Cobifurcating branches of solutions for nonlinear eigenvalue problems in Banach spaces, Ann. Mat. Pura Appl. 135 (4) (1983), 119–131.Google Scholar
9.Granas, A. and Dugundji, J., Fixed point theory, Springer Monographs in Mathematics (Springer, New York, 2003).CrossRefGoogle Scholar
10.Krasnoselskii, M. A., Topological methods in the theory of nonlinear integral equations (Macmillan, New York, 1964).Google Scholar
11.Prodi, G. and Ambrosetti, A., Analisi non lineare, Scuola Normale Superiore Pisa, Quaderno 1 (1973).Google Scholar
12.Rabinowitz, P. H., Minimax methods in critical point theory with applications to differential equations, CBMS Regional Conference Series Math., vol. 65 (AMS, Providence, RI, 1986).CrossRefGoogle Scholar
13.Stuart, C. A., An introduction to bifurcation theory based on differential calculus, in Nonlinear analysis and mechanics: Heriot-Watt Symposium, Vol. 4 (Res. Notes Math. vol. 39; Pitman, London, 1979), 76–135.Google Scholar