Summary: Based on the idea of partial Yosida approximation, we prove global in time existence of solutions for nonhomogeneous initial-boundary value problems for the class of coercive models of monotone type from the theory of inelastic deformations of metals. Then, using this result and the coercive limit idea, we approximate self-controlling problems with nonhomogeneous boundary data by a sequence of coercive models, and prove a convergence result.