- Tytuł:
- On uniformly approximate convex vector-valued function
- Autorzy:
- Rolewicz, S.
- Powiązania:
- https://bibliotekanauki.pl/articles/206344.pdf
- Data publikacji:
- 2012
- Wydawca:
- Polska Akademia Nauk. Instytut Badań Systemowych PAN
- Tematy:
-
vector valued functions
strongly α(·)-K-paraconvexity
differentiable manifolds
Gateaux and Frechet differentiability - Opis:
- Let X, Y be real Banach spaces. Let Z be a Banach space partially ordered by a pointed closed convex cone K. Let f(·) be a locally uniformly approximate convex function mapping an open subset ΩY ⊂ Y into Z. Let ΩX ⊂ X be an open subset. Let σ(·) be a differentiable mapping of ΩX into ΩY such that the differentials of σ/x are locally uniformly continuous function of x. Then f(σ(·)) mapping X into Z is also a locally uniformly approximate convex function. Therefore, in the case of Z = Rn the composed function f(σ(·)) is Frechet differentiable on a dense Gδ-set, provided X is an Asplund space, and Gateaux differentiable on a dense Gδ-set, provided X is separable. As a consequence, we obtain that in the case of Z = Rn a locally uniformly approximate convex function defined on a C1,uE -manifold is Frechet differentiable on a dense Gδ-set, provided E is an Asplund space, and Gateaux differentiable on a dense Gδ-set, provided E is separable.
- Źródło:
-
Control and Cybernetics; 2012, 41, 2; 443-462
0324-8569 - Pojawia się w:
- Control and Cybernetics
- Dostawca treści:
- Biblioteka Nauki
Artykuł