Skip to main content
SpringerLink
Log in

On volterra equations of the first kind

  • Published:
Integral Equations and Operator Theory Aims and scope Submit manuscript

Abstract

The existence of a solution β of the equation

$$\int_0^t {a(t - s)d\beta (s) = 1, t > 0} $$

is studied under fairly general assumptions on the function a. Sufficient conditions for the measure β to be absolutely continuous or satisfy some additional regularity properties are given. An extension to nonconvolution kernels is also considered.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Barbu V.: Nonlinear semigroups and differential equations in Banach spaces. Noordhoff International Publishing, Leyden, 1976.

    Google Scholar 

  2. Gripenberg G.: On positive, nonincreasing resolvents of Volterra equations. J. Differential Equations 30 (1978), 380 - 390.

    Article  Google Scholar 

  3. Gripenberg G.: On the resolvents of nonconvolution Volterra kernels. Funkcial. Ekvac., to appear.

  4. Gripenberg G.: An abstract nonlinear Volterra equation. Israel J. Math. 34 (1979), 198 - 212.

    Google Scholar 

  5. Hirschman I.I. and Widder D.V.: The convolution transform. Princeton University Press, Princeton, 1955.

    Google Scholar 

  6. Pogorzelski W.: Integral equations and their applications, Vol. I. Pergamon Press, Oxford, 1966.

    Google Scholar 

  7. Shea D.F. and Wainger S.: Variants of the Wiener-Lévy theorem, with applications to stability problems for some Volterra integral equations. Amer. J. Math. 97 (1975), 312 - 343.

    Google Scholar 

  8. Shohat J.A. and Tamarkin J.D.: The problem of moments. American Mathematical Society, Providence, 1943.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics, Helsinki University of Technology, SF-02150, Espoo 15, Finland

    Gustaf Gripenberg

Authors
  1. Gustaf Gripenberg
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gripenberg, G. On volterra equations of the first kind. Integr equ oper theory 3, 473–488 (1980). https://doi.org/10.1007/BF01702311

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01702311

Keywords

Search

Navigation