Nonlocal spectral problem for a second-order differential equation with an involution

Abstract

For the spectral problem −u^''(x) + αu^''(−x) = λu(x), −1 < x < 1, with nonlocal boundary conditions
u(−1) = βu(1), u^'(−1) = u^'(1), where α ∈ (−1, 1), β²<>1, we study the spectral properties. We show
that if r =√(1 − α)/(1 + α) is irrational, then the system of eigenfunctions is complete and minimal in L₂(−1, 1) but is not a basis. In the case of a rational number r, the root subspace of the problem consists of eigenvectors and an infinite number of associated vectors. In this case, we indicated a method for choosing associated functions that provides the system of root functions of the problem is an unconditional basis in L₂(−1, 1).