Bounded solutions of linear differential equations in a banach space

For a linear inhomogeneous differential equation in a Banach space, we find a criterion for the existence of solutions that are bounded on the entire real axis under the assumption that the homogeneous equation admits an exponential dichotomy on the semiaxes. This result is a generalization of the Palmer lemma to the case of infinite-dimensional spaces. We consider examples of countable systems of ordinary differential equations that have bounded solutions. __________ Translated from Neliniini Kolyvannya, Vol. 9, No. 1, pp. 3–14, January–March, 2006.