Linear Volterra-Stieltjes integral equations in the sense of the Kurzweil-Henstock integral

Márcia Federson; Ricardo Bianconi

Archivum Mathematicum (2001)

  • Volume: 037, Issue: 4, page 307-328
  • ISSN: 0044-8753

Abstract

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In 1990, Hönig proved that the linear Volterra integral equation x t - ( K ) ∫ a , t α t , s x s d s = f t , t ∈ a , b , where the functions are Banach space-valued and f is a Kurzweil integrable function defined on a compact interval a , b of the real line ℝ , admits one and only one solution in the space of the Kurzweil integrable functions with resolvent given by the Neumann series. In the present paper, we extend Hönig’s result to the linear Volterra-Stieltjes integral equation x t - ( K ) ∫ a , t α t , s x s d g s = f t , t ∈ a , b , in a real-valued context.

How to cite

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Federson, Márcia, and Bianconi, Ricardo. "Linear Volterra-Stieltjes integral equations in the sense of the Kurzweil-Henstock integral." Archivum Mathematicum 037.4 (2001): 307-328. <http://eudml.org/doc/248739>.

@article{Federson2001,
abstract = {In 1990, Hönig proved that the linear Volterra integral equation \[ x\left( t\right) -\,(K)\int \nolimits \_\{\left[ a,t\right] \}\alpha \left( t,s\right) x\left( s\right)\,ds=f\left( t\right)\,,\qquad t\in \left[ a,b\right]\,, \] where the functions are Banach space-valued and $f$ is a Kurzweil integrable function defined on a compact interval $\left[ a,b\right] $ of the real line $\mathbb \{R\}$, admits one and only one solution in the space of the Kurzweil integrable functions with resolvent given by the Neumann series. In the present paper, we extend Hönig’s result to the linear Volterra-Stieltjes integral equation \[ x\left( t\right) - (K)\int \nolimits \_\{\left[ a,t\right] \}\alpha \left( t,s\right) x\left( s\right) dg\left( s\right) =f\left( t\right) ,\qquad t\in \left[ a,b\right]\,, \] in a real-valued context.},
author = {Federson, Márcia, Bianconi, Ricardo},
journal = {Archivum Mathematicum},
keywords = {linear integral equations; Kurzweil-Henstock integrals; linear integral equations; Kurzweil-Henstock integrals},
language = {eng},
number = {4},
pages = {307-328},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Linear Volterra-Stieltjes integral equations in the sense of the Kurzweil-Henstock integral},
url = {http://eudml.org/doc/248739},
volume = {037},
year = {2001},
}

TY - JOUR
AU - Federson, Márcia
AU - Bianconi, Ricardo
TI - Linear Volterra-Stieltjes integral equations in the sense of the Kurzweil-Henstock integral
JO - Archivum Mathematicum
PY - 2001
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 037
IS - 4
SP - 307
EP - 328
AB - In 1990, Hönig proved that the linear Volterra integral equation \[ x\left( t\right) -\,(K)\int \nolimits _{\left[ a,t\right] }\alpha \left( t,s\right) x\left( s\right)\,ds=f\left( t\right)\,,\qquad t\in \left[ a,b\right]\,, \] where the functions are Banach space-valued and $f$ is a Kurzweil integrable function defined on a compact interval $\left[ a,b\right] $ of the real line $\mathbb {R}$, admits one and only one solution in the space of the Kurzweil integrable functions with resolvent given by the Neumann series. In the present paper, we extend Hönig’s result to the linear Volterra-Stieltjes integral equation \[ x\left( t\right) - (K)\int \nolimits _{\left[ a,t\right] }\alpha \left( t,s\right) x\left( s\right) dg\left( s\right) =f\left( t\right) ,\qquad t\in \left[ a,b\right]\,, \] in a real-valued context.
LA - eng
KW - linear integral equations; Kurzweil-Henstock integrals; linear integral equations; Kurzweil-Henstock integrals
UR - http://eudml.org/doc/248739
ER -

References

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