ISSN 2074-1863 Ufa mathematical journal. Volume 3. № 2 (2011). Pp. 27-32. UDC 517.927.25 STABILITY OF BASIS PROPERTY OF A TYPE OF PROBLEMS ON EIGENVALUES WITH NONLOCAL PERTURBATION OF BOUNDARY CONDITIONS N.S. IMANBAEV, M.A. SADYBEKOV Abstract. The article is devoted to a spectral problem for a multiple differentiation operator with an integral perturbation of boundary conditions of one type which are regular, but not strongly regular. The unperturbed problem has an asymptotically simple spectrum, and its system of normalized eigenfunctions creates the Riesz basis. We construct the characteristic determinant of the spectral problem with an integral perturbation of the boundary conditions. The perturbed problem can have any finite number of multiple eigenvalues. Therefore, its root subspaces consist of its eigen and (maybe) adjoint functions. It is shown that the Riesz basis property of a system of eigen and adjoint functions is stable with respect to integral perturbations of the boundary condition. Keywords: Riesz basis, regular boundary conditions, eigenvalues, root functions, spectral problem, integral perturbation of boundary condition, characteristic determinant 1. Problem statement It is well known that a system of eigenfunctions of the operator, given by a formally self-adjoint differential expression with arbitrary self-adjoint boundary conditions providing a discrete spectrum, form an orthonormal basis of the space L 2 . The problem of preserving the basis properties with a (weak, in a sense ) perturbation of the initial operator was investigated in many works. For example, the similar question for the case of a self-adjoint, and nonself-adjoint initial operator was investigated in [1 - 3], and in [4 - 6], respectively. The present paper is devoted to the spectral problem l(u) ≡-u 00 (x)= λu(x), 0 <x< 1, (1) U 1 (u) ≡ u 0 (0) - u 0 (1) + αu(0) = 0, (2) U 2 (u) ≡ u(0) - u(1) = Z 1 0 p(x)u(x)dx, p(x) ∈ L 2 (0, 1), (3) which is close to investigations [3, 5]. Here α 6=0 is an arbitrary complex number. In [3], the stability of basis properties of a periodic problem (the case α =0) for Equation (1) was studied with an integral perturbation of the boundary condition. It was proved that the set P of functions p(x), that provide the problem by the basis property of eigenfunctions, is dense in L 1 (0, 1), the set L 1 (0, 1)\P is also dense in L 1 (0, 1). The problem on basis property of the root functions of the operator with more general integral boundary conditions is solved positively in [5], where the Riesz basis property with brackets is proved under the Birkhoff regularity condition [7, p. 66–67] of the boundary conditions of the unperturbed problem; and the Riesz basis property is proved under the auxiliary assumption of strengthened regularity. In our case, the unperturbed boundary conditions (2), (3) (when c  Imanbaev N.S., Sadybekov M.A. 2011. Submitted on 25 March 2011. 27