# Implicit function theorem without a priori assumptions about normality

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## Abstract

The equation *F*(*x*, σ) = 0,*x* ∈ *K*, in which σ 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*_{*}, σ_{*}) 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*.

## Keywords

implicit function theory 2-regularity condition Robinson condition convex cone## Preview

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