Solvability of Nonlinear Equations in Case of Branching Solutions

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-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 are illustrated on the nonlinear integral equations.

Construction of solutions of integral equations with Stieltjes functionals and bifurcation parameters

“…In paper [Sidorovm Sidorov (2021)] the problem statement for the integral equation with single load is given. Then, in [Sidorov, Dreglea Sidorov (2023), Sidorov, Dreglea Sidorov (2022)] theory of the Hammerstein integral equations with loads and bifurcation parameters was proposed. In [Lampe, Rosenwasser (2010)] the Fredholm resolvent was employed for computing H 2 -norm for linear periodic systems.…”