On implicit function theorems at abnormal points

We consider the equation $F(x,\sigma)=0$, $x\in K$, in which $\sigma$ is a parameter and $x$ is an unknown variable taking values in a specified convex cone $K$ lying in a Banach space $X$. This equation is investigated in a neighborhood of a given solution $(x_*,\sigma_*)$, where Robinson's constraint qualification may be violated. We introduce the 2-regularity condition, which is considerably weaker than Robinson's constraint qualification; assuming that it is satisfied, we obtain an implicit function theorem for this equation. The theorem is a generalization of the known implicit function theorems even in the case when the cone $K$ coincides with the whole space $X$.