Differential equations of fourth order on infinite interval

In the present paper we consider the Dirichlet problem for the differential-operator equation of the fourth order where , is a linear opeator in the separable Hilbert space and has a complete system of the eigenfunctins, which form a Riesz basis in We prove the existence and uniqueness of the generalized solution for the Dirichlet problem and give the description of the spectrum for the corresponding operator.

Keywords:

Dirichlet problem , weighted Sobolev spaces , differential equations in abstract spaces , spectrum of the linear operator