Unconditional bases related to a nonclassical second-order differential operator

In the present paper, we introduce the notion of regularity of boundary conditions for a simplest second-order differential equation with a deviating argument. We prove the Riesz basis property for a system of root vectors of the corresponding generalized spectral problem with regular boundary conditions (in the sense of the introduced definition). Examples of irregular boundary conditions to which the theory of Il’in basis property can be applied are given.