Mathematical Notes, vol. 78, no. 4, 2005, pp. 573–576. Translated from Matematicheskie Zametki, vol. 78, no. 4, 2005, pp. 619–621. Original Russian Text Copyright c 2005 by A. V. Arutyunov. BRIEF COMMUNICATIONS Implicit-Function Theorem on the Cone in a Neighborhood of an Irregular Point A. V. Arutyunov Received February 16, 2005 Key words: implicit-function theorem, Banach space, 2-regular mapping, convex cone. Suppose that X is a topological space, Y and Z are Banach spaces, and K ⊆ Y is a closed convex cone. Suppose we are also given a mapping F : X × Y → Z and points x 0 ∈ X , y 0 ∈ Y such that F (x 0 ,y 0 )=0. Consider the following system consisting of an equation and an inclusion: F (x, y)=0 , y ∈ K, (1) where y is an unknown quantity and x is a parameter. Our goal is as follows: under the assumption that the mapping F is smooth in a neighborhood of the point (x 0 ,y 0 ) , to obtain for (1) an implicit- function theorem without a priori assuming that the Robinson regularity condition ∂F ∂y (x 0 ,y 0 )(K + Lin{y 0 })= Z is satisfied. (Here Lin is the linear hull of a set.) The theorems presented in this paper generalize certain results from the paper [1], which contains a bibliography on implicit-function and inverse- function theorems without a priori regularity assumptions. We assume that the mapping F is twice continuously differentiable with respect to y in a neighborhood of the point (x 0 ,y 0 ) and, for each fixed x , its second partial derivative (∂ 2 F/∂y 2 )(x, · ) satisfies the Lipschitz condition in the variable y with the same Lipschitz constant independent of x . The mappings F ( · ,y 0 ) , ∂F ∂y ( · ,y 0 ) , ∂ 2 F ∂y 2 ( · ,y 0 ) are assumed continuous at the point x 0 . Let K = K + Lin{y 0 } , C = ∂F ∂y (x 0 ,y 0 )(K) and assume that the relative interior ri C of the cone C is not empty, its linear hull Lin C is closed and this subspace can be topologically complemented. By π we denote the linear continuous operator projecting Z onto some subspace complementing Lin C . Definition (see [1]). Suppose that h ∈K , ∂F ∂y (x 0 ,y 0 )h =0 , − ∂ 2 F ∂y 2 (x 0 ,y 0 )[h, h] ∈ ri C. (2) A mapping F is said to be 2 -regular at a point (x 0 ,y 0 ) with respect to K along the direction h if Lin  ∂F ∂y (x 0 ,y 0 )(K)  + ∂ 2 F ∂y 2 (x 0 ,y 0 )  h, K∩ Ker ∂F ∂y (x 0 ,y 0 )  = Z. 0001-4346/2005/7834-0573 c 2005 Springer Science+Business Media, Inc. 573