Boundary Value Problems for Ordinary Differential Equations with Linear Dependence on the Spectral Parameter

The paper considers boundary value problems generated by an ordinary differential expression of the nth order and arbitrary boundary conditions with linear dependence on the spectral parameter both in the equation and the boundary conditions. Classes of problems are defined, which are called regular and strongly regular. Linear operators in the space $$H = {{L}_{2}}[0,1] \oplus {{\mathbb{C}}^{m}},\;m \leqslant n,$$ are assigned to these problems, and the corresponding adjoint operators are constructed in explicit form. In the general form, we solve the problem of selecting “superfluous” eigenfunctions, which was previously studied only for the special cases of second- and fourth-order equations. Namely, a criterion is found for selecting m eigen- or associated (root) functions of a regular problem so that the remaining system of root functions forms a Riesz basis or a Riesz basis with parenthesis in the original space $${{L}_{2}}[0,1]$$ .