A Class of Second Order Tangent Sets

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ASS OF SECOND ORDER TANGENT SETS S. S. Kutateladze

UDC 517.972.8

Abstract: Under consideration are the construction and properties of some special class of second other tangent sets on using the technique of nonstandard analysis. DOI: 10.1134/S0037446620050079 Keywords: second order tangent, Clarke cone, Nelson internal set theory

Let X be a real vector space. Assume that we are given some almost vector topology σ with the zero neighborhood ﬁlter Nσ := σ(0) as well as some almost vector topology τ with the ﬁlter Nτ := τ (0). Recall that every almost vector topology σ on X is characterized by the two properties: Firstly, multiplication by each scalar is continuous; and, secondly, addition is jointly continuous. It is clear that X admits an almost vector topology σ such that σ(0) coincides with a ﬁxed ﬁlter N if and only if the monad μ(N ) is an external vector space over the external ﬁeld of standard scalars. In the sequel, σ will be a vector topology, unless stated otherwise explicitly. It is comfortable to work in the assumption of standard environment within Nelson internal set theory IST (see [1]). Recall that the monad μ(F ) of a standard ﬁlter F is the external intersection of the standard elements of F . As usual, introduce the inﬁnite proximity that is associated with the appropriate uniformity in X; i. e., x1 ≈σ x2 ↔ x1 − x2 ∈ μ(Nσ ). Note that the monad μσ (x) of the neighborhood ﬁlter σ(x) of the topology σ is as follows: μσ (x) := x + μ(Nσ ). Let ≈ stand for the inﬁnite proximity on the reals R. Recall that if given are some subset F of X and some point x ¯ in X, then subdiﬀerential calculus (see [2]) deals in particular with the Hadamard, Clarke, and Bouligand cones F − x ; intτ Ha(F, x ¯ ) := λ U ∈σ(¯ x) λ>0

Cl(F, x ¯ ) :=

x ∈F ∩U 00 00

x∈F ∩U 0 0, λ ≈ 0} is the conatus of v (see [1, Subsection 5.1.2]). Recall (see, for instance, [7]) that 1 ¯ , v) := h ∈ X : (∀λn ↓ 0)(∃hn → h) x ¯ + λn v + λ2n hn ∈ F . A(2) (F, x 2 To simplify bulky formulas we will assume that f is continuous at x ¯ ∈ F with respect to the topology τ on X. Theorem 1. The following holds:

=

Cl(2) (F, x ¯ )(v1 , v2 ) F − x − λ v − λ v

V ∈Nσ U ∈τ (¯ x) x ∈F ∩U v ∈ F −x ∩V 1 λ V1 ∈σ(v1 ) λ1 >0 00 0 0, λ > 0) ↔ (∀x ≈τ x (∃v1 ≈σ v1 )(∃v2 ≈σ v2 )(∃v ≈σ v) x + λ v1 ∈ F ∧ x + λ v2 ∈ F ∧ x + λ v1 + λ v2 + 4λ λ v ∈ F. Denote the set on the right-hand side of the claim by A. Take v ∈ Cl(2) (F, x ¯ )(v1 , v2 ) and some standard neighborhoods V ∈ Nσ , V1 ∈ σ(v1 ), and V2 ∈ σ(v2 ). If λ1 and λ2 are strictly positive inﬁnitesimal 845

while U is an inﬁnitesimal τ -neighborhood of x ¯ ; i. e., U ⊂ μτ (¯ x); then there are some v1 ≈σ v1 , v2 ≈σ v2 , and v ≈σ v such that x + λ v1 ∈ F , x + λ v2 ∈ F , and x + λ v1 + λ v2 + 4λ λ v ∈ F for all x ∈ F ∩ U , 0 < λ ≤ λ1 , and 0 < λ ≤ λ2 . In other words, there exist v1 ∈ (F − x )/λ ∩ V1 , v2 ∈ (F − x )/λ ∩ V2 , and v ∈ v + V satisfying the needed properties. Since the parameters are standa

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