Global Surjection and Global Inverse Mapping Theorems in Banach Spaces

It is well known that a nonlinear operator that is one-to-one locally (i.e., in a neighborhood of any point a t which it is defined) may fail to be one-to-one on the entire domain, even if the latter is very regular (say, a convex body; see reference 1, p. 25). Global inverse mapping theorems, therefore, are of considerable interest to us. We refer to reference 1 for a detailed discussion that is mainly concerned with the finite dimensional situation and its applications. It is probably in reference 2 that a general theorem for C’-maps in Banach spaces was first proved. The assumption of the theorem is a combination of the standard criterion for local univalence and a condition of a global nature. The second principal result of this paper has a similar structure, but it deals with arbitrary continuous mappings and uses a slightly weakened form of the global condition (in this part, however, without changing essentially the techniques developed by Plastock’). This theorem is accompanied by a number of local univalence criteria for nondifferentiable maps. The first result that we prove here is a global surjection theorem that offers a lower estimate for the image of a map with a closed graph (which is a guaranteed radius of a ball contained within the image). As with the inverse map theorem of which we spoke of above, this one uses a combination of a local sufficient condition (this time for surjection) and a global condition similar to that in the other theorem. We refer to references 3 and 4 for specific local surjection criteria for nondifferentiable maps. In what follows, X and Yare Banach spaces, and F is a map from the whole of X into Y.