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Nonlocal spectral problem for a second-order differential equation with an involution 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), β 2 = 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).

The spectral problem −u ′′ (x) + αu ′′ (−x) = λu(x) , −1 < x < 1 , with nonlocal boundary conditions u(−1) = βu(1) , u ′ (−1) = u ′ (1) , is studied in the spaces Lp(−1, 1) for any α ∈ (−1, 1) and β ̸ = ±1. It is proved that if r = √ (1 − α)/(1 + α) is irrational then the system of its eigenfunctions is complete and minimal in Lp(−1, 1) for any p > 1 , but does not form a basis. In the case of a rational value of r , the way of supplying this system with associated functions is specified to make all the root functions a basis in Lp(−1, 1) .