On uniformly approximate convex vector-valued function

• Autorzy: Rolewicz S.
• Języki publikacji: EN

Abstrakty

EN

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 Ω_Y ⊂ Y into Z. Let Ω_X ⊂ X be an open subset. Let σ(·) be a differentiable mapping of Ω_X into Ω_Y 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 C^1,u_E -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.

Słowa kluczowe

EN

vector valued functions; strongly α(·)-K-paraconvexity; differentiable manifolds; Gateaux and Frechet differentiability

• Wydawca: Systems Research Institute, Polish Academy of Sciences
• Czasopismo: Control and Cybernetics
• Rocznik: 2012
• Tom: Vol. 41, no. 2
• Strony: 443--462
• Opis fizyczny: Bibliogr. 35 poz.

Twórcy

autor

Rolewicz S.

• Institute of Mathematics of the Polish Academy of Sciences Śniadeckich 8, 00-956 Warszawa,

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