A nonexistence result for the Kurzweil integral

Pavel Krejčí; Jaroslav Kurzweil

Mathematica Bohemica (2002)

  • Volume: 127, Issue: 4, page 571-580
  • ISSN: 0862-7959

Abstract

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It is shown that there exist a continuous function f and a regulated function g defined on the interval [ 0 , 1 ] such that g vanishes everywhere except for a countable set, and the K * -integral of f with respect to g does not exist. The problem was motivated by extensions of evolution variational inequalities to the space of regulated functions.

How to cite

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Krejčí, Pavel, and Kurzweil, Jaroslav. "A nonexistence result for the Kurzweil integral." Mathematica Bohemica 127.4 (2002): 571-580. <http://eudml.org/doc/249016>.

@article{Krejčí2002,
abstract = {It is shown that there exist a continuous function $f$ and a regulated function $g$ defined on the interval $[0,1]$ such that $g$ vanishes everywhere except for a countable set, and the $K^*$-integral of $f$ with respect to $g$ does not exist. The problem was motivated by extensions of evolution variational inequalities to the space of regulated functions.},
author = {Krejčí, Pavel, Kurzweil, Jaroslav},
journal = {Mathematica Bohemica},
keywords = {Kurzweil integral; regulated functions; Kurzweil integral; regulated functions; evolution variational inequalities},
language = {eng},
number = {4},
pages = {571-580},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A nonexistence result for the Kurzweil integral},
url = {http://eudml.org/doc/249016},
volume = {127},
year = {2002},
}

TY - JOUR
AU - Krejčí, Pavel
AU - Kurzweil, Jaroslav
TI - A nonexistence result for the Kurzweil integral
JO - Mathematica Bohemica
PY - 2002
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 127
IS - 4
SP - 571
EP - 580
AB - It is shown that there exist a continuous function $f$ and a regulated function $g$ defined on the interval $[0,1]$ such that $g$ vanishes everywhere except for a countable set, and the $K^*$-integral of $f$ with respect to $g$ does not exist. The problem was motivated by extensions of evolution variational inequalities to the space of regulated functions.
LA - eng
KW - Kurzweil integral; regulated functions; Kurzweil integral; regulated functions; evolution variational inequalities
UR - http://eudml.org/doc/249016
ER -

References

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  2. Regulated functions, Math. Bohem. 119 (1991), 20–59. (1991) MR1100424
  3. Introduction to the theory of integration, Academic Press, New York, 1963. (1963) Zbl0112.28302MR0154957
  4. Generalized variational inequalities, J. Convex Anal. 9 (2002), 159–183. (2002) MR1917394
  5. Generalized ordinary differential equations and continuous dependence on a parameter, Czechoslovak Math. J. 7 (1957), 418–449. (1957) Zbl0090.30002MR0111875
  6. On the relation between Young’s and Kurzweil’s concept of Stieltjes integral, Časopis Pěst. Mat. 98 (1973), 237–251. (1973) Zbl0266.26006MR0322113
  7. On a modified sum integral of Stieltjes type, Časopis Pěst. Mat. 98 (1973), 274–277. (1973) Zbl0266.26007MR0322114
  8. Differential and Integral Equations: Boundary Value Problems and Adjoints, Academia and D. Reidel, Praha, 1979. (1979) MR0542283
  9. Regulated functions and the Perron-Stieltjes integral, Časopis Pěst. Mat. 114 (1989), 187–209. (1989) MR1063765

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