scholarly journals A Remark on Isometries of Absolutely Continuous Spaces

2020 ◽  
Vol 2020 ◽  
pp. 1-3
Author(s):  
Alireza Ranjbar-Motlagh
Keyword(s):  
Continuous Function ◽  
Composition Operator ◽  
Real Line ◽  
Weighted Composition Operator ◽  
Function Spaces ◽  
Absolutely Continuous ◽  
The Real ◽  
Absolutely Continuous Function ◽  
Weighted Composition ◽  
Vector Valued

The purpose of this article is to study the isometries between vector-valued absolutely continuous function spaces, over compact subsets of the real line. Indeed, under certain conditions, it is shown that such isometries can be represented as a weighted composition operator.

scholarly journals Representation of Group Isomorphisms: The Compact Case

2015 ◽  
Vol 2015 ◽  
pp. 1-6
Author(s):  
Marita Ferrer ◽  
Margarita Gary ◽  
Salvador Hernández
Keyword(s):  
Continuous Function ◽  
Composition Operator ◽  
Discrete Group ◽  
Weighted Composition Operator ◽  
Continuous Functions ◽  
Group Isomorphism ◽  
Compact Spaces ◽  
Mild Conditions ◽  
Weighted Composition ◽  
Compact Case

LetGbe a discrete group and letAandBbe two subgroups ofG-valued continuous functions defined on two 0-dimensional compact spacesXandY. A group isomorphismHdefined betweenAandBis calledseparatingwhen, for each pair of mapsf, g∈Asatisfying thatf-1eG∪g-1eG=X, it holds thatHf-1eG∪Hg-1eG=Y. We prove that under some mild conditions every biseparating isomorphismH:A→Bcan be represented by means of a continuous functionh:Y→Xas a weighted composition operator. As a consequence we establish the equivalence of two subgroups of continuous functions if there is a biseparating isomorphism defined between them.

A new method of proof of Filippov’s theorem based on the viability theorem

2012 ◽  
Vol 10 (6) ◽  
Author(s):  
Sławomir Plaskacz ◽  
Magdalena Wiśniewska
Keyword(s):  
Continuous Function ◽  
Differential Inclusion ◽  
New Method ◽  
Initial Condition ◽  
Contingent Derivative ◽  
Absolutely Continuous ◽  
Absolutely Continuous Function ◽  
Gamma S

AbstractFilippov’s theorem implies that, given an absolutely continuous function y: [t 0; T] → ℝd and a set-valued map F(t, x) measurable in t and l(t)-Lipschitz in x, for any initial condition x 0, there exists a solution x(·) to the differential inclusion x′(t) ∈ F(t, x(t)) starting from x 0 at the time t 0 and satisfying the estimation $$\left| {x(t) - y(t)} \right| \leqslant r(t) = \left| {x_0 - y(t_0 )} \right|e^{\int_{t_0 }^t {l(s)ds} } + \int_{t_0 }^t \gamma (s)e^{\int_s^t {l(\tau )d\tau } } ds,$$ where the function γ(·) is the estimation of dist(y′(t), F(t, y(t))) ≤ γ(t). Setting P(t) = {x ∈ ℝn: |x −y(t)| ≤ r(t)}, we may formulate the conclusion in Filippov’s theorem as x(t) ∈ P(t). We calculate the contingent derivative DP(t, x)(1) and verify the tangential condition F(t, x) ∩ DP(t, x)(1) ≠ ø. It allows to obtain Filippov’s theorem from a viability result for tubes.

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