Linear Stieltjes integral equations in Banach spaces. II. Operator valued solutions

Štefan Schwabik

Mathematica Bohemica (2000)

  • Volume: 125, Issue: 4, page 431-454
  • ISSN: 0862-7959

Abstract

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This paper is a continuation of [9]. In [9] results concerning equations of the form x(t) = x(a) +at [A(s)]x(s) +f(t) - f(a) were presented. The Kurzweil type Stieltjes integration in the setting of [6] for Banach space valued functions was used. Here we consider operator valued solutions of the homogeneous problem (t) = I +dt [A(s)](s) as well as the variation-of-constants formula for the former equation.

How to cite

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Schwabik, Štefan. "Linear Stieltjes integral equations in Banach spaces. II. Operator valued solutions." Mathematica Bohemica 125.4 (2000): 431-454. <http://eudml.org/doc/248663>.

@article{Schwabik2000,
abstract = {This paper is a continuation of [9]. In [9] results concerning equations of the form x(t) = x(a) +at [A(s)]x(s) +f(t) - f(a) were presented. The Kurzweil type Stieltjes integration in the setting of [6] for Banach space valued functions was used. Here we consider operator valued solutions of the homogeneous problem (t) = I +dt [A(s)](s) as well as the variation-of-constants formula for the former equation.},
author = {Schwabik, Štefan},
journal = {Mathematica Bohemica},
keywords = {linear Stieltjes integral equations; generalized linear differential equation; equation in Banach space; linear Stieltjes integral equations; generalized linear differential equation; equation in Banach space},
language = {eng},
number = {4},
pages = {431-454},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Linear Stieltjes integral equations in Banach spaces. II. Operator valued solutions},
url = {http://eudml.org/doc/248663},
volume = {125},
year = {2000},
}

TY - JOUR
AU - Schwabik, Štefan
TI - Linear Stieltjes integral equations in Banach spaces. II. Operator valued solutions
JO - Mathematica Bohemica
PY - 2000
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 125
IS - 4
SP - 431
EP - 454
AB - This paper is a continuation of [9]. In [9] results concerning equations of the form x(t) = x(a) +at [A(s)]x(s) +f(t) - f(a) were presented. The Kurzweil type Stieltjes integration in the setting of [6] for Banach space valued functions was used. Here we consider operator valued solutions of the homogeneous problem (t) = I +dt [A(s)](s) as well as the variation-of-constants formula for the former equation.
LA - eng
KW - linear Stieltjes integral equations; generalized linear differential equation; equation in Banach space; linear Stieltjes integral equations; generalized linear differential equation; equation in Banach space
UR - http://eudml.org/doc/248663
ER -

References

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  1. Ju. L. Daletskij M. G. Krejn, Stability of Solutions of Differential Equations in Banach Spaces, Nauka, Moskva, 1970. (In Russian.) (1970) MR0352638
  2. N. Dunford J. T Schwartz, Linear Operators I, Interscience Publishers, New York, 1958. (1958) MR0117523
  3. Ch. S. Hönig, Volterra-Stieltjes Integral Equations, North-Holland Publ. Comp., Amsterdam, 1975. (1975) MR0499969
  4. J. Kurzweil, Nichtabsolut konvergente Integrale, B. G. Teubner Verlagsgesellschaft, Leipzig, 1980. (1980) Zbl0441.28001MR0597703
  5. W. Rudin, Functional Analysis, McGraw-Hill Book Company, New York, 1973. (1973) Zbl0253.46001MR0365062
  6. Š. Schwabik, Abstract Perron-Stieltjes integral, Math. Bohem. 121 (1996), 425-447. (1996) Zbl0879.28021MR1428144
  7. Š. Schwabik, Generalized Ordinary Differential Equations, World Scientific, Singapore, 1992. (1992) Zbl0781.34003MR1200241
  8. Š. Schwabik M. Tvrdý O. Vejvoda, Differential and Integral Equations, Academia & Reidel, Praha & Dordrecht, 1979. (1979) MR0542283
  9. Š. Schwabik, Linear Stieltjes integral equations in Banach spaces, Math. Bohem. 124 (1999), 433-457. (1999) MR1722877

Citations in EuDML Documents

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  1. Štefan Schwabik, Operator-valued functions of bounded semivariation and convolutions
  2. Štefan Schwabik, A note on integration by parts for abstract Perron-Stieltjes integrals
  3. Giselle A. Monteiro, Milan Tvrdý, On Kurzweil-Stieltjes integral in a Banach space
  4. Umi Mahnuna Hanung, Milan Tvrdý, On the relationships between Stieltjes type integrals of Young, Dushnik and Kurzweil
  5. Rodolfo Collegari, Márcia Federson, Miguel Frasson, Linear FDEs in the frame of generalized ODEs: variation-of-constants formula

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