Abstract
It is known that the system of eigenfunctions of the classical biharmonic operator with the Dirichlet boundary condition is complete and orthonormal in space. The eigenvalues corresponding to these eigenfunctions are positive and can be numbered in ascending order. In some cases, eigenfunctions and eigenvalues of boundary value problems for a perturbed biharmonic operator have similar properties. In this work, a nonlocal analogue of the perturbed biharmonic operator is introduced using orthogonal matrices. Spectral problems of two boundary value problems are studied for this operator. Dirichlet boundary conditions are considered in the first problem, and Dirichlet-type conditions in the second. When studying the first problem, we use the complete system of eigenfunctions of the Dirichlet problem for the perturbed biharmonic operator. Using the properties of these systems, as well as the properties of the image with orthogonal matrices, we find the eigenfunctions and eigenvalues of the main problem. In the second task, we use the eigenfunctions and eigenvalues of the Dirichlet problem for the Laplace operator. Using the explicit form, as well as the properties of these systems, we construct the eigenfunctions and eigenvalues of the second task. The proof of the theorem on the completeness of the system of eigenfunctions of the considered tasks in space is provided.