Bounded solutions of linear difference equations of higher orders in a Banach space

Abstract

We study the problem of the existence of a unique bounded solution of a linear difference equation of the $p$th order with bounded operator coefficients. We consider the case where the $p$th order operator matrix corresponding to a difference equation is similar to a diagonal operator matrix on the main diagonal of which the roots of the “algebraic” operator equation corresponding to a given difference equation are located.