Some problems of spectral theory of fourth-order differential operators with regular boundary conditions

In this paper, we consider the problem yIV+q(x)y=λy,0<x<1,y″′(1)-(-1)σy″′(0)+αy′(0)+γy(0)=0,y″(1)-(-1)σy″(0)+βy(0)=0,y′(1)-(-1)σy′(0)=0,y(1)-(-1)σy(0)=0where λ is a spectral parameter; q(x) ∈ L1(0 , 1) is a complex-valued function; α, β, γ 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 and it is proved that all the eigenvalues, except for a finite number, are simple in the case αβ≠ 0. It is shown that the system of root functions of this spectral problem forms a basis in the space Lp(0 , 1) , 1 < p< ∞, when αβ≠ 0 ; moreover, this basis is unconditional for p = 2. [MediaObject not available: see fulltext.] © 2014, The Author(s).