We present a class of generalized Nash equilibrium problems (GNEP) in Banach space. Issues concerning the existence of equilibria and constraint qualifications will be discussed. Using a class of convex penalty functions, we reduce the question of solving the GNEP to solving a sequence of approximations in the form of classical Nash equilibrium problems. Finally, we apply the theoretical results to a class of problems whose constraint sets are partially governed by a linear parabolic partial differential equation. An algorithm for this class of problems will be presented and illustrated by a few examples.