Abstract
A high-order linear equation in a Banach space with a bounded operator \(A\) at the unknown function is studied. The equation is considered on \(\mathbb{R}\) without initial conditions. A theorem on the existence and uniqueness of a solution to the equation is proved using the two-sided Laplace transform in a specially defined spaces of continuously differentiable and exponentially bounded functions. The inverse Laplace transform of the resolvent of operator \(A\) plays a key role in the representation of the solution to the equation. Abstract results are applied to the study of high-order linear system of ordinary differential equations and boundary value problems for high-order in time equations with polynomials of a self-adjoint elliptic differential operator with respect to spatial variables.
