Mappings sharing a value on finite-dimensional spaces

Sucient conditions are given to assert that di€erentiable mappings and di€eren-tiable proper mappings between R n and R m share a value. The proof of the result is constructive and is based upon continuation methods. 1. Preliminaries In this paper priori conditions are given to prove that a function has at least one zero. Other sucient conditions have been given in previous papers [1±4] to guarantee the existence of zero points. These conditions were essentially compactness conditions. In paper [5], we studied the case of bounded perturbations of proper mappings. Here, we study the existence of zeros of di€er-entiable perturbations of proper mappings between ®nite dimensional Banach spaces of not necessarily the same dimension. To prove our theorem we use continuation methods (see [1±14]). They supply the existence of curves reaching zero points [1±4,6,10,15,16]. Essentially, we have to study a suitable homotopy proving its zeros are in a compact set.