On Stability of Continuous Extensions of Mappings with Respect to Nemytskii Operator
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On Stability of Continuous Extensions of Mappings with Respect to Nemytskii Operator A. V. Arutyunova,* and S. E. Zhukovskiya,** Presented by Academician of the RAS A.T. Fomenko January 23, 2020 Received February 6, 2020; revised February 6, 2020; accepted February 15, 2020
Abstract—The concept of stability of continuous extension of mappings with respect to a Nemytskii superposition operator is studied. Sufficient conditions of such stability with respect to a Nemytskii superposition operator are obtained. The essentiality of the corresponding assumptions is illustrated by examples. Keywords: Nemytskii operator, stability, theorems on extensions of continuous mappings DOI: 10.1134/S1064562420030047
Let X be a Banach space with norm ⋅ X , (Y , ρY ) be a metric space with metric ρY, Σ be a Hausdorff paracompact topological space, and f : X × Σ → Y be a given continuous mapping. Let C be a given nonempty closed subset of Σ. For a continuous mapping ϕ: Σ → X , its restriction to C is denoted by ϕC. The Nemytskii (superposition) operator 1 is defined as 1(ϕ)(σ) = f (ϕ(σ), σ). It takes the continuous mapping ϕ: Σ → X to a continuous mapping N (ϕ): Σ → Y . For continuous mappings ϕ: C → X , the operator 1 is defined in a similar manner. The usual norm in the space of continuous bounded mappings ϕ: Σ → X is denoted by ||⋅||, and the same notation ||⋅|| is used to denote the norm for continuous bounded mappings from C to X. The usual metric in the space of continuous bounded mappings from Σ to Y is denoted by ρ, and the same notation ρ is used to denote the metric for continuous bounded mappings from C to Y. The same notation used for the norm and metric of mappings defined on Σ and C does not lead to confusion. By the theorem on a continuous extension, which follows from the Michael selection theorem (see [1; 2, Section 1.4]), an arbitrary continuous mapping defined on C can be continuously extended to Σ . Accordingly, the following question arises: does there exist an a Trapeznikov
Institute of Control Sciences, Russian Academy of Sciences, Moscow, 117997 Russia *e-mail: [email protected] **e-mail: [email protected]
extension that is stable with respect to the Nemytskii operator? More specifically, given a continuous mapping ϕ0: Σ → X , is it true that, for any ε > 0, there exists δ > 0 such that, for any continuous mapping ϕ: C → X satisfying
||ϕ − ϕC0 || < δ,
ρ(1(ϕ), 1(ϕC0 )) < δ,
(1)
ˆ of ϕ such that there exists a continuous extension ϕ ˆ − ϕ0 || < ε, ||ϕ
ˆ ), 1(ϕ0)) < ε ? ρ(1(ϕ
(2)
This problem is of interest in itself and arises in the problem of extending an implicit function defined on a closed subset C ⊂ Σ . This can be explained as follows. Consider an equation f ( x, σ) = 0 for the unknown x ∈ X and a parameter σ ∈ Σ. For each value of σ, the task is to solve this equation for x , i.e., to find a continuous mapping ϕ: Σ → X such that f (ϕ(σ), σ) = 0 ∀σ ∈ Σ . The last means that 1(ϕ) = 0. The mapping ϕ is called an implicit function. Assuming that X and Y are Hilbert spaces, the mapping f s
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