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Article

Solvability and Bifurcation of Solutions of Nonlinear Equations with Fredholm Operator

1
Institute of Mathematics and Information Technologies, Irkutsk State University, 664003 Irkutsk, Russia
2
Energy Systems Institute, Russian Academy of Sciences, 664033 Irkutsk, Russia
3
Baikal School of BRICS, Irkutsk National Research Technical University, 664003 Irkutsk, Russia
*
Author to whom correspondence should be addressed.
Symmetry 2020, 12(6), 912; https://doi.org/10.3390/sym12060912
Received: 30 March 2020 / Revised: 6 May 2020 / Accepted: 7 May 2020 / Published: 1 June 2020
The necessary and sufficient conditions of existence of the nonlinear operator equations’ branches of solutions in the neighbourhood of branching points are derived. The approach is based on the reduction of the nonlinear operator equations to finite-dimensional problems. Methods of nonlinear functional analysis, integral equations, spectral theory based on index of Kronecker-Poincaré, Morse-Conley index, power geometry and other methods are employed. Proposed methodology enables justification of the theorems on existence of bifurcation points and bifurcation sets in the nonstandard models. Formulated theorems are constructive. For a certain smoothness of the nonlinear operator, the asymptotic behaviour of the solutions is analysed in the neighbourhood of the branch points and uniformly converging iterative schemes with a choice of the uniformization parameter enables the comprehensive analysis of the problems details. General theorems and effectiveness of the proposed methods are illustrated on the nonlinear integral equations. View Full-Text
Keywords: branch points; bifurcation points; Fredholm operator; uniformization; asymptotics; iterations; regularization branch points; bifurcation points; Fredholm operator; uniformization; asymptotics; iterations; regularization
MDPI and ACS Style

Sidorov, N.; Sidorov, D.; Dreglea, A. Solvability and Bifurcation of Solutions of Nonlinear Equations with Fredholm Operator. Symmetry 2020, 12, 912. https://doi.org/10.3390/sym12060912

AMA Style

Sidorov N, Sidorov D, Dreglea A. Solvability and Bifurcation of Solutions of Nonlinear Equations with Fredholm Operator. Symmetry. 2020; 12(6):912. https://doi.org/10.3390/sym12060912

Chicago/Turabian Style

Sidorov, Nikolai, Denis Sidorov, and Aliona Dreglea. 2020. "Solvability and Bifurcation of Solutions of Nonlinear Equations with Fredholm Operator" Symmetry 12, no. 6: 912. https://doi.org/10.3390/sym12060912

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