Geodesic
PERSISTENCE OF THE NORMALIZED EIGENVECTORS OF A PERTURBED OPERATOR IN THE VARIATIONAL CASE
Glasgow mathematical journal, Tome 55 (2013) no. 3, pp. 629-638

Voir la notice de l'article provenant de la source Cambridge University Press

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.
DOI : 10.1017/S0017089512000791
Mots-clés : Primary 47A55, Secondary 47J05, 47J10, 47J15 
CHIAPPINELLI, RAFFAELE; FURI, MASSIMO; PERA, MARIA PATRIZIA. PERSISTENCE OF THE NORMALIZED EIGENVECTORS OF A PERTURBED OPERATOR IN THE VARIATIONAL CASE. Glasgow mathematical journal, Tome 55 (2013) no. 3, pp. 629-638. doi: 10.1017/S0017089512000791
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     title = {PERSISTENCE {OF} {THE} {NORMALIZED} {EIGENVECTORS} {OF} {A} {PERTURBED} {OPERATOR} {IN} {THE} {VARIATIONAL} {CASE}},
     journal = {Glasgow mathematical journal},
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[1] 1.Berger, M. S., Nonlinearity and functional analysis (Academic Press, New York, 1977). Google Scholar

[2] 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. Google Scholar | DOI

[3] 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] 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] 5.Crandall, M. G. and Rabinowitz, P. H., Bifurcation from simple eigenvalues, J. Funct. Anal. 8 (1971), 321–340. Google Scholar

[6] 6.Deimling, K., Nonlinear functional analysis (Springer, New York, 1985). Google Scholar

[7] 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] 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] 9.Granas, A. and Dugundji, J., Fixed point theory, Springer Monographs in Mathematics (Springer, New York, 2003). Google Scholar | DOI

[10] 10.Krasnoselskii, M. A., Topological methods in the theory of nonlinear integral equations (Macmillan, New York, 1964). Google Scholar

[11] 11.Prodi, G. and Ambrosetti, A., Analisi non lineare, Scuola Normale Superiore Pisa, Quaderno 1 (1973). Google Scholar

[12] 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). Google Scholar | DOI

[13] 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

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