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Change of variables formula under minimal assumptions

Piotr Hajłasz (1993)

Colloquium Mathematicae

In the previous papers concerning the change of variables formula (in the form involving the Banach indicatrix) various assumptions were made about the corresponding transformation (see e.g. [BI], [GR], [F], [RR]). The full treatment of the case of continuous transformation is given in [RR]. In [BI] the transformation was assumed to be continuous, a.e. differentiable and with locally integrable Jacobian. In this paper we show that none of these assumptions is necessary (Theorem 2). We only need...

Characterization of compact subsets of curves with ω-continuous derivatives

Marcin Pilipczuk (2011)

Fundamenta Mathematicae

We give a characterization of compact subsets of finite unions of disjoint finite-length curves in ℝⁿ with ω-continuous derivative and without self-intersections. Intuitively, our condition can be formulated as follows: there exists a finite set of regular curves covering a compact set K iff every triple of points of K behaves like a triple of points of a regular curve. This work was inspired by theorems by Jones, Okikiolu, Schul and others that characterize compact subsets of...

Characterization of convex functions

Jacek Tabor, Józef Tabor (2009)

Studia Mathematica

There are many inequalities which in the class of continuous functions are equivalent to convexity (for example the Jensen inequality and the Hermite-Hadamard inequalities). We show that this is not a coincidence: every nontrivial linear inequality which is valid for all convex functions is valid only for convex functions.

Characterization of Globally Lipschitz Nemytskiĭ Operators Between Spaces of Set-Valued Functions of Bounded φ-Variation in the Sense of Riesz

N. Merentes, J. L. Sánchez Hernández (2004)

Bulletin of the Polish Academy of Sciences. Mathematics

Let (X,∥·∥) and (Y,∥·∥) be two normed spaces and K be a convex cone in X. Let CC(Y) be the family of all non-empty convex compact subsets of Y. We consider the Nemytskiĭ operators, i.e. the composition operators defined by (Nu)(t) = H(t,u(t)), where H is a given set-valued function. It is shown that if the operator N maps the space R V φ ₁ ( [ a , b ] ; K ) into R W φ ₂ ( [ a , b ] ; C C ( Y ) ) (both are spaces of functions of bounded φ-variation in the sense of Riesz), and if it is globally Lipschitz, then it has to be of the form H(t,u(t)) = A(t)u(t)...

Characterization of intermediate values of the triangle inequality II

Hiroki Sano, Tamotsu Izumida, Ken-Ichi Mitani, Tomoyoshi Ohwada, Kichi-Suke Saito (2014)

Open Mathematics

In [Mineno K., Nakamura Y., Ohwada T., Characterization of the intermediate values of the triangle inequality, Math. Inequal. Appl., 2012, 15(4), 1019–1035] there was established a norm inequality which characterizes all intermediate values of the triangle inequality, i.e. C n that satisfy 0 ≤ C n ≤ Σj=1n ‖x j‖ − ‖Σj=1n x j‖, x 1,...,x n ∈ X. Here we study when this norm inequality attains equality in strictly convex Banach spaces.

Characterization of ω -limit sets of continuous maps of the circle

David Pokluda (2002)

Commentationes Mathematicae Universitatis Carolinae

In this paper we extend results of Blokh, Bruckner, Humke and Sm’ıtal [Trans. Amer. Math. Soc. 348 (1996), 1357–1372] about characterization of ω -limit sets from the class 𝒞 ( I , I ) of continuous maps of the interval to the class 𝒞 ( 𝕊 , 𝕊 ) of continuous maps of the circle. Among others we give geometric characterization of ω -limit sets and then we prove that the family of ω -limit sets is closed with respect to the Hausdorff metric.

Characterizations of Kurzweil-Henstock-Pettis integrable functions

L. Di Piazza, K. Musiał (2006)

Studia Mathematica

We prove that several results of Talagrand proved for the Pettis integral also hold for the Kurzweil-Henstock-Pettis integral. In particular the Kurzweil-Henstock-Pettis integrability can be characterized by cores of the functions and by properties of suitable operators defined by integrands.

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