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Solvability of Nonlinear Equations in Case of Branching Solutions
Version 1
: Received: 16 April 2020 / Approved: 17 April 2020 / Online: 17 April 2020 (01:34:37 CEST)
A peer-reviewed article of this Preprint also exists.
A. Sidorov, N.; Sidorov, D.; Dreglea, A.I. Solvability and Bifurcation of Solutions of Nonlinear Equations with Fredholm Operator. Symmetry 2020, 12, 912. A. Sidorov, N.; Sidorov, D.; Dreglea, A.I. Solvability and Bifurcation of Solutions of Nonlinear Equations with Fredholm Operator. Symmetry 2020, 12, 912.
Abstract
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 reduction of the nonlinear operator equations to finite-dimensional problems. Methods of nonlinear functional analysis, integral equations, spectral theory based on index of Kronecker-Poincare, 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 are illustrated on the nonlinear integral equations.
Keywords
branch points; bifurcation points; Fredholm operator; uniformization; asymptotics; iterations; regularization
Subject
Computer Science and Mathematics, Analysis
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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