Summary: 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 \(K\) 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 \(X\).