Friction is a phenomenon which is present in most mechanical devices and frequently encountered in everyday life. Noncoercive incremental contact problems with Coulomb friction constitute an important class of friction problems due to their frequent occurrence in mechanical engineering. The topic of this thesis is to study this class of friction problems, and we obtain conditions for the existence of solutions. The thesis is divided in two parts:

1. The first part focuses on discrete noncoercive linearly elastic systems, and a sufficient condition for the existence of a solution is derived. This condition is a compatibility condition on the applied force, which can be interpreted as a stability condition for a rigid body in contact with the obstacle.
2. The second part focuses on a two-dimensional continuous elastic system in contact with a plane obstacle. A non-local model of friction is used. The result is a compatibility condition on the applied force, which is similar to the corresponding discrete setting.