Abstract. Conditions for existence of solutions bounded on $R$ for a linear and nonlinear ordinary differential system are obtained under the assumption that the operator $L$ defined by the corresponding unperturbed linearized homogeneous system is of Fredholm type. If $L$ is a Fredholm operator with index zero or $L$ is a Fredholm operator and, in addition, has an exponential trichotomy on $R$, we obtain the well-known results.