Continuous selections and invertibility of nonsmooth maps between Banach spaces

Abstract

In the setting of Banach spaces, we address the problems of local surjectivity, existence of a continuous selection and invertibility for nonsmooth maps which admit a suitable pseudo-Jacobian. We obtain a general sufficient condition for the existence of a continuous selection. As a consequence, we recover from our result the classical Bartle-Graves selection theorem for linear maps. We also obtain a sufficient condition for local invertibility, which in particular applies to certain Gâteaux differentiable maps.