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The completion of the space of convex, bounded sets with respect to the Demyanov metric

Grzybowski J., Leśniewski A., Rzeżuchowski T.

Demonstratio Mathematica

2013

Vol. 46, nr 1

191--196

The Demyanov metric in the family of convex, compact sets in finite dimensional spaces has been recently extended to the family of convex, bounded sets – not necessarily closed. In this note it is shown that these spaces are not complete and a model for the completion is proposed. A full answer is given in R2 and the situation in higher dimensions is discussed.

The Demyanov metric and measurable multifunctions

Leśniewski A.

Journal of Mathematics and Applications

2010

Vol. 33

61-65

We present in this paper measurability multifunctions in the family of all convex, bounded sets which need not be closed. The Demyanov metric is discussed.

Compact and bounded sets in intuitionistic fuzzy metric spaces

Efe H., Alaca C.

Demonstratio Mathematica

2007

Vol. 40, nr 2

449-456

The purpose of this paper is to define q-bounded, semi-bounded, totally bounded, and unbounded sets in an intuitionistic fuzzy metric space X and study the relation between F-bounded sets and the above mentioned sets and prove that the statements (a) X is compact (b) X is sequentially compact and (c) X is complete and totally bounded are all equivalent in an intuitionistic fuzzy metric space X.

Abstract Boundedness and the Stability of the Pexider Equation

Ger R.

Scientific Issues of Jan Długosz University in Częstochowa. Mathematics

2003

Vol. 9

41--49

Let (S, +) be a semigroup (not necessarily Abelian) and let (X,+) be a commutative group. We deal with an axiomatically given family B ⸦ 2x of "bounded sets" and with mappings f, g, h : S → X such that the transformation S x S ∋ (x, y) → f (x + y) - g (x) - h (y) ∈ X remains B-bounded. Stability results existing in the literature in connection with the Pexider functional equation become special cases of our theorems up to the magnitude of approximating constants.