Spectral Features of the Solving of a Fredholm Homogeneous Integro-Differential Equation with Integral Conditions and Reflecting Deviation

Abstract

The problems of solvability and construction of solutions of a nonlocal boundary value problem for the second-order Fredholm integro-differential equation with degenerate kernel, integral conditions, spectral parameters and reflecting deviation are considered. Using the method of the degenerate kernel, the boundary value problem is integrated as an ordinary differential equation. When we define the arbitrary integration constants there are possible five cases with respect to the first spectral parameter. Calculated values of the spectral parameter for each case. Further, the problem is reduced to solving systems of linear algebraic equations. Irregular values of the second spectral parameter are determined. At irregular values of the second spectral parameter the Fredholm determinant is degenerate. Other values of the second spectral parameter, for which the Fredholm determinant does not degenerate, are called regular values. Taking the values of the first spectral parameter into account for regular values of the second spectral parameter the corresponding solutions were constructed for each of five cases. The stability of the solution of the boundary value problem for given values in integral conditions is proved. The conditions under which the solution of the boundary value problem will be small are studied. For the irregular values of the second spectral parameter each of the five cases is checked separately. The orthogonality conditions are used. Cases are determined in which the problem has an infinite number of solutions and these solutions are constructed. For other cases, the absence of nontrivial solutions of the problem is proved.