Bounded solutions of weakly nonlinear differential equations in a Banach space

For a weakly nonlinear differential equation in a Banach space, we establish necessary and sufficient conditions for the existence of solutions bounded on the entire real axis under the assumption that the generating equation has bounded solutions and the corresponding homogeneous equation admits an exponential dichotomy on the semiaxes. __________ Translated from Neliniini Kolyvannya, Vol. 11, No. 2, pp. 151–159, April–June, 2008.