Global surjection and global inverse mapping theorems in Banach spaces.

*(English)*Zbl 0709.47054
Rep. Moscow Refusnik Semin., Ann. N. Y. Acad. Sci. 491, 181-188 (1987).

Summary: [For the entire collection see Zbl 0705.00008.]

It is well known that a nonlinear operator that is one-to-one locally may fail to be one-to-one on the entire domain, even if the latter is very regular. Global inverse mapping theorems, therefore, are of considerable interest to us. It is probably in R. Plastock [Trans. Am. Math. Soc. 200, 169-183 (1974; Zbl 0291.54009)] that a general theorem for \(C^ 1\)-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, loc. cit.). This theorem is accompanied by a number of local univalence criterion 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.

It is well known that a nonlinear operator that is one-to-one locally may fail to be one-to-one on the entire domain, even if the latter is very regular. Global inverse mapping theorems, therefore, are of considerable interest to us. It is probably in R. Plastock [Trans. Am. Math. Soc. 200, 169-183 (1974; Zbl 0291.54009)] that a general theorem for \(C^ 1\)-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, loc. cit.). This theorem is accompanied by a number of local univalence criterion 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.

##### MSC:

47J05 | Equations involving nonlinear operators (general) |