Bochner product integration

Štefan Schwabik

Mathematica Bohemica (1994)

  • Volume: 119, Issue: 3, page 305-335
  • ISSN: 0862-7959

Abstract

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A new definition of the product integral is given. The definition is based on a procedure which is analogous to the sum definition of the Bochner integral given by J. Kurzweil and E.J. McShane. The new definition is shown to be equivalent to the seemingly verey different one given by J.D. Dollard and C.N. Friedman in [1] and [2].

How to cite

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Schwabik, Štefan. "Bochner product integration." Mathematica Bohemica 119.3 (1994): 305-335. <http://eudml.org/doc/29316>.

@article{Schwabik1994,
abstract = {A new definition of the product integral is given. The definition is based on a procedure which is analogous to the sum definition of the Bochner integral given by J. Kurzweil and E.J. McShane. The new definition is shown to be equivalent to the seemingly verey different one given by J.D. Dollard and C.N. Friedman in [1] and [2].},
author = {Schwabik, Štefan},
journal = {Mathematica Bohemica},
keywords = {Kurzweil-Henstock integral; Bochner integral; product integral; Bochner product integral; Kurzweil-Henstock integral; Bochner integral; product integral},
language = {eng},
number = {3},
pages = {305-335},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Bochner product integration},
url = {http://eudml.org/doc/29316},
volume = {119},
year = {1994},
}

TY - JOUR
AU - Schwabik, Štefan
TI - Bochner product integration
JO - Mathematica Bohemica
PY - 1994
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 119
IS - 3
SP - 305
EP - 335
AB - A new definition of the product integral is given. The definition is based on a procedure which is analogous to the sum definition of the Bochner integral given by J. Kurzweil and E.J. McShane. The new definition is shown to be equivalent to the seemingly verey different one given by J.D. Dollard and C.N. Friedman in [1] and [2].
LA - eng
KW - Kurzweil-Henstock integral; Bochner integral; product integral; Bochner product integral; Kurzweil-Henstock integral; Bochner integral; product integral
UR - http://eudml.org/doc/29316
ER -

References

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  1. J. D. Dollard C.N. Friedman, Product Integration with Applications to Differential Equations, Addison-Wesley Publ. Company, Reading, Massachusetts, 1979. (1979) MR0552941
  2. J. D. Dollard C.N. Friedman, 10.1016/0022-1236(78)90091-5, J. Func. Anal. 28 (1978), 309-354. (1978) MR0492656DOI10.1016/0022-1236(78)90091-5
  3. N. Dunford J.T. Schwartz, Linear Operators I., Interscience Publishers, New York, London, 1958. (1958) MR0117523
  4. R. D. Gill S. Johansen, 10.1214/aos/1176347865, Annals of Statistics 18 (1990), no. 4, 1501-1555. (1990) MR1074422DOI10.1214/aos/1176347865
  5. R. Henstock, Lectures on the Theory of Integration, World Scientific, Singapore, 1988. (1988) Zbl0668.28001MR0963249
  6. R. Henstock, The General Theory of Integration, Clarendon Press, Oxfoгd, 1991. (1991) Zbl0745.26006MR1134656
  7. E. Hille R. S. Phillips, Functional Analysis and Semi-groups, American Mathematical Society, Providence, 1957. (1957) MR0089373
  8. J. Jarník J. Kurzweil, Perron integral, Perron product integral and ordinary linear differential equations, EQUADIFF 6, Lecture Notes in Mathematics 1192, Springer, Berlin (1986), 149-154. (1986) MR0877117
  9. J. Jarník J. Kurzweil, A general form of the product integral and linear ordinary differential equations, Czech. Math. Јournal 37 (1987), 642-659. (1987) MR0913996
  10. J. Kurzweil, Nichtabsolut konvergente Integrale, BSB B. G. Teubner Verlagsgesellschaft, Leipzig, 1980. (1980) Zbl0441.28001MR0597703
  11. J. Mawhin, Analyse. Fondements,techniques,évolution, De Boeck-Wesmael, Bruxelles, 1992. (1992) Zbl0759.26004MR1190926
  12. E. J. McShane, Unified Integration, Academic Press, New York, London, 1983. (1983) Zbl0551.28001MR0740710
  13. L. Schlesinger, Vorlesungen über lineare Differentialgleichungen, Teubner, Leipzig, 1908. (1908) 
  14. L. Schlesinger, 10.1007/BF01174342, Math. Zeitschrift 33 (1931), 33-61. (1931) Zbl0001.01503DOI10.1007/BF01174342
  15. G. Schmidt, On multiplicative Lebesgue integration and families of evolution operators, Math. Scand. 29 (1971), 113-133. (1971) Zbl0232.47024MR0352398
  16. Š. Schwabik, The Perron product integral and generalized linear differential equations, Časopis pěst. mat. 115 (1990), 368-404. (1990) Zbl0724.26006MR1090861
  17. Š. Schwabik, Generalized Ordinary Differential Equations, World Scientific, Singapore, 1992. (1992) Zbl0781.34003MR1200241
  18. V. Volterra, Sulle equazioni differenziali lineari, Rend. Accademia dei Lincei 3 (1887), 393-396. 
  19. V. Volterra B. Hostinský, Operations infinitesimales linéaires, Gauthier-Villars, Paris, 1938. (1938) 
  20. K. Yosida, Functional Analysis, Springer, Berlin, 1965. (1965) Zbl0126.11504

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