This paper generalizes some results of Gale and Debreu related to results of Bohnen-blust and Karlin, commonly used to prove the existence of equilibrium in competitive economies. Applied to economics, these results allow further latitude in the choice of admissible price vectors, and in the conditions defining equilibrium. These results are also applicable in more general contexts where, instead of prices, one considers a set of controls or instruments of policy and where, instead of equilibrium, another goal is sought. These goals are related to the appropriate instruments through the concept of dual convex cones. Some examples are provided.