Bifurcation theory of loaded integral equations

The study focuses on analyzing the nonlinear Hammerstein and linear Fredholm integral equations, considering both local and nonlocal functionals. The research aims to determine necessary and sufficient conditions (existence theorems) for the coefficients of the equation and identify the parameter values (bifurcation points) in its neighborhood where nontrivial real solutions exist. The dominant terms of the solutions’ asymptotic behavior are determined, and methods for constructively solving both regular and irregular cases of the linear Fredholm integral functional equations of the second kind are proposed. The solutions are expressed as Taylor series for regular cases and as Laurent series for irregular cases. Several examples are provided to demonstrate the applicability of the constructive existence theorems.