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On the Riemann integral of set-valued functions

Piątek B.

Zeszyty Naukowe. Matematyka Stosowana / Politechnika Śląska

2012

z. 2

5--18

In the paper we present a generalization of the Riemann integral of set-valued function introduced by Dinghas and independently by Hukuhara to the case of nonempty bounded closed convex subsets of Banach spaces. Moreover, we compare properties of this integral to the Riemann integral of real function and consider its relation with the Hukuhara derivative.

Praca zawiera uogólnienie definicji całki Riemanna z funkcji wielowartościowej wprowadzonej przez Dinghasa i niezależnie przez Hukuharę na przypadek niepustych domkniętych, ograniczonych i wypukłych podzbiorów przestrzeni Banacha. Następnie porównujemy własności tak zdefiniowanej całki z całką Riemanna z funkcji rzeczywistej oraz rozważamy jej związek z pochodną Hukuhary.

On the continuity of the integrable multifunctions

Piątek B.

Opuscula Mathematica

2009

Vol. 29, no. 1

81-88

The generalization of the Polovinkin theorem is studied.

Teachers studies students difficulties concerning the generalization of the concept of the Riemann integral

Powązka Z., Zaręba L.

Scientific Issues of Jan Długosz University in Częstochowa. Mathematics

2007

Vol. 12

347-354

One of the most fundamental concepts of the mathematical analysis is the Riemann integral. For a teacher of mathematics the concept of the integral is important because of the connections with the Jordan measure which is considered in the elementary geometry. Besides the Riemann integral the course of mathematical analysis includes multiple integrals, line integrals and surface integrals. In this paper we present the results of our research concerning the difficulties of students in noticing mutual connections between different kinds of integrals.

On the Sincov functional equation

Piątek B.

Demonstratio Mathematica

2005

Vol. 38, nr 4

875--881

The Riemarm integral, the Hukuhara differences and derivatives will be applied to obtain solutions of the Sincov functional equation.

On fuzzy integral equations

Friedman M., Ming M., Kandel A.

Fundamenta Informaticae

1999

Vol. 37, Nr 1,2

89-99

Using the embedding method, the fuzzy integral is represented as a parametric Riemann inte-gral. An algorithm which approximates this integral uniformly, is incorporated to design a soft computing tool for solving a fuzzy Fredholm integral equation of the second kind with arbi-trary kernel, using a uniformly convergent iterative procedure.