On Riescz bases of eigenfunction of $2$-nd order differential operator with involution and integral boundary conditions

Riesz basisness with brackets of the eigen and associated function is proved for a $2$-nd order differential operator with involution in the derivatives and with integral boundary conditions. To demonstrate this the spectral problem of the initial operator is reduced to the spectral problem of a $1$-st order operator without involution in the $4$-dimensional vector-function space. The equation of the new spectral problem contains a difficult non-trivial coefficient of the unknown function, but after a transformation, depending on the spectral parameter $\lambda$, this coefficient can be estimated as $O(\lambda^{-1/2})$. This makes it possible to get under some regularity conditions the location of eigenvalues of the initial operator and to present its resolvent by integral operators of simpler structure. These facts together with completeness of the eigen and associated functions of the operator, adjoint to the initial one, underlie the proof of the result formulated.