Geodesic
On analytic perturbations of linear equations in the case of incomplete generalized Jordan set
Contemporary Mathematics. Fundamental Directions, Science — Technology — Education — Mathematics — Medicine, Tome 68 (2022) no. 1, pp. 80-94 
D. G. Rakhimov; D. Akhmadzhanova. On analytic perturbations of linear equations in the case of incomplete generalized Jordan set. Contemporary Mathematics. Fundamental Directions, Science — Technology — Education — Mathematics — Medicine, Tome 68 (2022) no. 1, pp. 80-94. http://geodesic.mathdoc.fr/item/CMFD_2022_68_1_a6/
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     title = {On analytic perturbations of linear equations in the case of incomplete generalized {Jordan} set},
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

Based on the methods of the theory of bifurcations, the problem of perturbation of linear equations by small analytic terms is considered. In contrast to the work of Trenogin [7], the case of an incomplete generalized Jordan set of a linear Fredholm operator acting from one Banach space to another Banach space is studied. A technique is proposed that uses the regularization of the Fredholm operator by a specially constructed finite-dimensional operator.

[1] Vainberg M. M., Trenogin V. A., Teoriya vetvleniya reshenii nelineinykh uravnenii, Nauka, M., 1969 | MR

[2] Loginov B. V., “K teorii vozmuschenii dlya neodnorodnykh uravnenii”, Issledovaniya po differentsialnym uravneniyam i ikh prilozheniyam, Ilym, Alma-Ata, 1965, 95–101

[3] Loginov B. V., Rusak Yu. B., “Obobschennaya zhordanova struktura v teorii vetvleniya”, Pryamye i obratnye zadachi dlya uravnenii s chastnymi proizvodnymi, Fan, Tashkent, 1978, 133–148 | MR

[4] Rakhimov D. G., “O vychislenii kratnykh sobstvennykh znachenii reduktsionnym metodom lozhnykh vozmuschenii”, Zhurn. Srednevolzh. mat. ob-va, 2010, no. 3, 106–112

[5] Rakhimov D. G., “O regulyarizatsii kratnykh sobstvennykh znachenii reduktsionnym metodom lozhnykh vozmuschenii”, Vestn. Samar. gos. un-ta. Estestv. ser., 2012, no. 6, 35–41 | MR | Zbl

[6] Rakhimov D. G., Loginov B. V., Vozmuscheniya v zadachakh na sobstvennye znacheniya, Tashkent. univ. inf. tekhn., Tashkent, 2020

[7] Trenogin V. A., “Lineinye uravneniya v prostranstve Banakha s malym parametrom”, Materialy 6 Mezhvuzovskoi fiz.-mat. nauch. konf. Dalnego Vostoka (Khabarovsk, 1967)

[8] Albrecht A., Howlett P., Verma G., “Inversion of operator pencils on Banach space using Jordan chains when the generalized resolvent has an isolated essential singularity”, Linear Algebra Appl., 595 (2020), 33–62 | DOI | MR | Zbl

[9] Albrecht A., Howlett P., Verma G., “Inversion of operator pencils on Hilbert space”, J. Aust. Math. Soc., 108:2 (2020), 145–176 | DOI | MR | Zbl

[10] Avrachenkov K. E., Filar J. A., Howlett P. G., Analytic perturbation theory and its applications, SIAM, Philadelphia, 2013 | MR | Zbl