On the Solvability of a Fourth-Order Operator-Differential Equation with Multiple Characteristic

Abstract

In the Sobolev-type space with exponential weight, we obtain sufficient conditions for the well-posed and unique solvability on the entire axis of a fourth-order operator-differential equation whose main part has a multiple characteristic. We establish estimates for the norms of the operators of intermediate derivatives related to the conditions of solvability. In addition, we deduce the relationship between the exponent of the weight and the lower bound of the spectrum of the main operator appearing in the principal part of the equation. The obtained results are illustrated by an example of a problem for partial differential equations.