Variational inequalities with a small parameter are considered, together with closely related nonlinear operator equations. The penalty method as applied to variational inequalities is studied, and Galerkin's method relative to nonlinear equations. Under conditions of monotonicity, coerciveness and hemicontinuity on the operators, uniform convergence (with respect to the small parameter) of the approximate solutions thus obtained to the exact solution is demonstrated. Bibliography: 12 titles.