A Regular Differential Operator with Perturbed Boundary Condition

M. A. Sadybekov ; N. S. Imanbaev

Matematičeskie zametki, Tome 101 (2017) no. 5, pp. 768-778

Résumé

The operator $\mathcal{L}_{0}$ generated by a linear ordinary differential expression of $n$th order and regular boundary conditions of general form is considered on a closed interval. The characteristic determinant of the spectral problem for the operator $\mathcal{L}_{1}$, where $\mathcal{L}_{1}$ is an operator with the integral perturbation of one of its boundary conditions, is constructed, assuming that the unperturbed operator $\mathcal{L}_{0}$ possesses a system of eigenfunctions and associated functions generating an unconditional basis in $L_{2}(0,1)$. Using the obtained formula, we derive conclusions about the stability or instability of the unconditional basis properties of the system of eigenfunctions and associated functions of the problem under an integral perturbation of the boundary condition. The Samarskii–Ionkin problem with integral perturbation of its boundary condition is used as an example of the application of the formula. \renewcommand{\qed}

Détail

Keywords: basis, regular boundary condition, eigenvalue, root function, spectral problem, integral perturbation of the boundary condition, characteristic determinant.