On tangent cone to systems of inequalities and equations in Banach spaces under relaxed constant rank condition

Abstract

Under the relaxed constant rank condition, introduced by Minchenko and Stakhovski, we prove that the linearized cone is contained in the tangent cone (Abadie condition) for sets represented as solution sets to systems of finite numbers of inequalities and equations given by continuously differentiable functions defined on Hilbert spaces.

On Lipschitz-like continuity of a class of set-valued mappings

“…In the finite dimensional case, where H = R s this fact has been proved in Theorem 1 of [24]. In the infinite-dimensional case considered in the present paper this fact has been proved in Theorem 6.3 of [2].…”

“…As already noted, if F satisfies RCRCQ at (p 0 , x 0 ), there are neighbourhoods V (p 0 ) and V (x 0 ) such that F satisfies RCRCQ at any point (p, x) where p ∈ V (p 0 ) ∩ domF , x ∈ F (p) ∩ V (x 0 ). Hence, the set F (p) satisfies RCRCQ at x and by Theorem 6.3 of [2], Γ(F (p), x) = T (F (p), x).…”

“…Lemma 2.1. ( [2,24]). Let the set-valued mapping F satisfy RCRCQ (relative to domF × H) at (p 0 , x 0 ) ∈ grF .…”