Advertisement

Acta Mathematicae Applicatae Sinica

, Volume 18, Issue 4, pp 553–560 | Cite as

Linear Volterra Integral Equations

  • M. FedersonEmail author
  • R. Bianconi
  • L. Barbanti
Original Papers

Abstract

The Kurzweil-Henstock integral formalism is applied to establish the existence of solutions to the linear integral equations of Volterra-type
$$ x{\left( t \right)} + \;{}^{ * }{\int_{{\left[ {a,t} \right]}} {\alpha {\left( s \right)}x{\left( s \right)}ds = f{\left( t \right)}} },\;t \in {\left[ {a,b} \right]}, $$
(1)

where the functions are Banach-space valued. Special theorems on existence of solutions concerning the Lebesgue integral setting are obtained. These sharpen earlier results.

Keywords

Linear Volterra integral equations Kurzweil-Henstock integrals 

2000 MR Subject Classification

45A05 26A39 28B05 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Barbanti, L. Linear Stieltjes equation with generalized Riemann integral and existence of regulated solutions. Acta Math. Appl. Sinica, 17(4): 526–531 (2001)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Federson, M., Bianconi, R. Linear integral equations of Volterra concerning the integral of Henstock. Real Anal. Exchange, 25(1): 389–417 (2000)zbMATHMathSciNetGoogle Scholar
  3. 3.
    Federson, M. Sobre a existêcia de soluções para Equações integrais Lineares com respeito a integrais de Gauge. Doctor Thesis, Institute of Mathematics and Statistics, University of São Paulo, Brazil (1998)Google Scholar
  4. 4.
    Gengian, L. On necessary conditions for Henstock integrability. Real Anal. Exchange, 18: 522–531 (1992– 93)MathSciNetGoogle Scholar
  5. 5.
    Hönig, C.S. Volterra-Stieltjes integral equations. In: Math. Studies, Vol.16, ed. by North-Holland Publ. Comp., Amsterdam (1975)Google Scholar
  6. 6.
    Hönig, C.S. A Riemaniann characterization of the Bochner-Lebesgue integral. Seminário Brasileiro de Análise, 35: 351–358 (1992)Google Scholar
  7. 7.
    Lee, P.Y. Lanzhou lectures on Henstock integration. World Sci., Singapore (1989)Google Scholar
  8. 8.
    McShane, E.J. A unified theory of integration. Am. Math. Monthly, 80: 349–359 (1973) zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Schechter, M. Principles of functional analysis. Academic Press, New York (1971)Google Scholar
  10. 10.
    Schwabik, S. Abstract Perron-Stieltjes integral. Math. Bohem., 121(4): 425–447 (1996)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of São PauloSão PauloBrazil
  2. 2.Department of MathematicsUniversity of São PauloSão PauloBrazil

Personalised recommendations