The existence of strict solutions to the initial value problem for the Boltzmann equation with force term is investigated. For a certain class of velocity dependent external forces compatible with equilibrium, and for Gra[dacute]s generalized hard molecular potentials, the global existence, uniqueness, positivity and approach to equilibrium of solutions to the spatially homogeneous Boltzmann equation is proved in the case of initial data close to equilibrium (in the sense that the first moments are those of the equilibrium solution and a certain norm of the deviation from equilibrium is small). A local existence theorem is stated in the spatially inhomogeneous case, both for Gradcar;s hard and soft intermolecular potentials. One of the main steps of the argument concerning the nonlinear part of the theory, is formalized in an abstract theorem of possible independent interest.