An implicit function theorem without a priori assumptions about normality

The equation $F(x,\sigma)=0$, $x\in K$, in which $\sigma$ is a parameter and $x$ is an unknown taking values in a given convex cone in a Banach space $X$, is considered. This equation is examined in a neighborhood of a given solution $(x^*,\sigma^*)$ for which the Robinson regularity condition may be violated. Under the assumption that the 2-regularity condition (defined in the paper), which is much weaker than the Robinson regularity condition, is satisfied, an implicit function theorem is obtained for this equation. This result is a generalization of the known implicit function theorems even for the case when the cone $K$ coincides with the entire space $X$.