On a Non-linear Open Mapping Theorem

Let c be a positive number.A continuous linear map F from a Banachspace X into a Banach space Y is said to be c-open if Y_1 F(X_c)where Y_1denotes the closed unit ball in Y and X_c the closed ball in X with centre at ori-gin and radius c.This is equivalent to require that the equation F_x=y has a so-lution x with ||x||≤||y|| whenever y∈Y. F is c-open for some c>0 if and only ifit is an open map.In this note we concern with a non-linear situation.Thus let