Spectral singularities of solutions to a boundary-value problem for the Fredholm integro-differential equation of the second order with reflection of argument

The problems of solvability and construction of solutions to a nonlocal boundary-value problem for the nonlinear second-order nonlinear Fredholm integro-differential equation with a degenerate kernel, integral conditions and reflection of the argument are considered. The method of the degenerate kernel for the Fredholm integral equation is applied and developed for the case of the second-order nonlinear integro-differential equation. The spectral values of the parameters are calculated and the features arising in solving systems of algebraic equations and in determining arbitrary constants are studied. Criteria for the unique solvability of the stated nonlinear problem for regular values of spectral parameters are established. The method of successive approximations and the method of contraction mappings are used. The continuity of the solution of a boundary-value problem with respect to integral data is shown. The condition of smallness of this solution is revealed. For the irregular values of the spectral parameters, the problems of the existence or nonexistence of solutions to the nonlocal boundary-value problem under consideration are studied and solutions of this problem in the case of existence are constructed.