In the present paper the method of boundary integral equations (BIE) is extended to dynamic inequality problems involving convex energy superpotentials, i.e. to problems involving monotone, possibly multivalued, relations between reactions and displacements, stress and strains, etc. Using semidiscretization with respect to time the authors obtain, within each time step, a minimum potential energy formulation, the equivalent variational inequality formulation and some equivalent saddle point formulations using appropriate Lagrangian functions. An elimination technique gives rise to minimum ‘principles’, on the boundary with respect to the unknown displacements or stresses of the time step under consideration; parameters are the velocities, etc., of the previous time step. It is also shown that these minimum problems are equivalent to multivalued boundary integral equation problems. The theory is illustrated by numerical examples, which also treat the case of impact of the structure with the support.