An inverse function theorem on a cone in the neighborhood of an abnormal point

Let X,Y be Banach spaces. Let F(x) be an operator mapping X into Y. We assume that F(x) is twice continuously differentiable. Let x_0 in X. Let Ksubset X be a closed convex cone. We denote by C=F'(x_0)(K) and by Y_1 the linear hull of C. par Let h in K. We say that the mapping F is 2-regular at the point x_0 in the direction h if Y_1 + F"(x_0)[h, {rm Ker},F'(x_0)cap K] =Y. par Having this notion the author gives sufficient conditions of second order for the existence of inverse or implicit functions