ON ROOT FUNCTIONS OF NONLOCAL DIFFERENTIAL SECOND-ORDER OPERATOR WITH BOUNDARY CONDITIONS OF PERIODIC TYPE

In this paper we consider one class of spectral problems for a nonlocal ordinary differential operator (with involution in the main part) with nonlocal boundary conditions of periodic type. Such problems arise when solving by the method of separation of variables for a nonlocal heat equation. We investigate spectral properties of the problem for the nonlocal ordinary differential equation Ly (x) equivalent to -y ''(x) + epsilon y '' (-x) = lambda y (x), < x < 1. Here lambda is a spectral parameter, vertical bar epsilon vertical bar < 1. Such equations are called nonlocal because they have a term y '' ( with involutional argument deviation. Boundary conditions are nonlocal y' (+ ay' (1) = 0, y (y (1) = 0. Earlier this problem has been investigated for the special case a =-1. We consider the case a not equal-1.. A criterion for simplicity of eigenvalues of the problem is proved: the eigenvalues will be simple if and only if the number r = root(1-epsilon) = (1 + epsilon) is irrational. We show that if the number r is irrational, then all the eigenvalues of the problem are simple, and the system of eigenfunctions of the problem is complete and minimal but does not form an unconditional basis in L-2( 1). For the case of rational numbers r, it is proved that a (chosen in a special way) system of eigen- and associated functions forms an unconditional basis in L-2( 1).