Spectral properties of some regular boundary value problems for fourth order differential operators

Abstract

In this paper we consider the problem y ıv + p2(x)y 00 + p1(x)y 0 + p0(x)y = λy, 0 < x < 1, y (s) (1) − (−1)σy (s) (0) +Xs−1 l=0 αs,ly (l) (0) = 0, s = 1, 2, 3, y(1) − (−1)σy(0) = 0, where λ is a spectral parameter; pj(x) ∈ L1(0, 1), j = 0, 1, 2, are complex-valued functions; αs,l, s = 1, 2, 3, l = 0, s − 1, are arbitrary complex constants; and σ = 0, 1. The boundary conditions of this problem are regular, but not strongly regular. Asymptotic formulae for eigenvalues and eigenfunctions of the considered boundary value problem are established in the case α3,2 + α1,0 =6 α2,1. It is proved that the system of root functions of this spectral problem forms a basis in the space Lp(0, 1), 1 < p < ∞, when α3,2 +α1,0 6= α2,1, pj(x) ∈ W j 1 (0, 1), j = 1, 2, and p0(x) ∈ L1(0, 1); moreover, this basis is unconditional for p = 2.