We present a theorem on the existence of a maximal element for a correspondence which is upper hemi-continuous in some variables and which satisfies with respect to the other ones one the following conditions : (i) lower semi-continuous if the space has a finite dimension, (ii) lower semi-continuous if the space is complete, (iii) open fibers. This theorem generalizes the result of Gale and Mas-Colell (1975-1979) and the one of Bergstrom (1975) and extend to the infinite dimensional setting the result of Gourdel (1995).