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Selections of bounded variation

Chistyakov V. V.

Journal of Applied Analysis

2004

Vol. 10, nr 1

1-82

The paper presents recent results concerning the problem of the existence of those selections, which preserve the properties of a given set-valued mapping of one real variable taking on compact values from a metric space. The properties considered are the boundedness of Jordan, essential or generalized variation, Lipschitz or absolute continuity. Selection theorems are obtained by virtue of a single compactness argument, which is the exact generalization of the Helly selection principle. For set-valued mappings with the above properties we obtain a Castaing-type representation and prove the existence of multivalued selections and selections which pass through the boundaries of the images of the set-valued mapping and which are nearest in variation to a given mapping. Multivalued Lipschitzian superposition operators acting on mappings of bounded generalized variation are characterized, and solutions of bounded generalized variation to functional inclusions and embeddings, including variable set-valued operators in the right hand side, are obtained. Bibliography contains 113 items.

Some properties of the Bézier-Durrmeyer type operators

Anioł G., Pych-Taberska P.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

2001

[Z] 41

1-11

For some classes of measurable locally bounded functions / on an interval /, the rate of pointwise convergence of the Bézier-Durrmeyer type modification of discrete Feller operators is estimated. In the main theorems the Chanturija modulus of variation is used.

Lipschitzian superposition operators between spaces of functions of bounded generalized variation with weight

Chistyakov V. V.

Journal of Applied Analysis

2000

Vol. 6, nr 2

173-186

We present some properties of real valued functions of bounded generalized variation of Riesz-Orlicz type including weight and characterize Lipschitzian superposition Nemytskii operators which map between spaces (in fact, Banach algebras) of these functions.