Linear differential second-order equations in Banach space and splitting of operators

We consider a linear differential second order equation in complex Banach space with bounded operator coefficients. We study the question of existence of bounded on the whole real axis solutions (with bounded right-hand side) and their asymptotic behaviour. The research is conducted in case of separated roots of the corresponding algebraic operator equation or providing that the norm of operator, placed in front of the first derivative in the equation, is small. In the latter case we apply the method of similar operators (operator splitting theorem). The main results are obtained with the use of the theorems on similarity transformation of second order operator matrix to a block-diagonal one.