DEGENERATE DIFFERENTIAL-OPERATOR EQUATIONS ON INFINITE INTERVAL

Abstract

In the present paper we consider the Dirichlet problem for the fourth order differential-operator equation $Lu \equiv(t^{\alpha}u^{\prime\prime})^{\prime\prime}+t^{-2}Au = f$ , where $t\in(1, +\infty), \alpha\geq 2, f \in L_{2,2}((1, +\infty), H),~A$ is a linear operator in the separable Hilbert space H and has a complete system of eigenvectors that form a Riesz basis in H. The existence and uniqueness of the generalized solution for the Dirichlet problem are proved, and the description of spectrum for the corresponding operator is given.