Skip to main content
SpringerLink
Log in

Existence of solutions of abstract volterra equations in a banach space and its subsets

  • Published:
Ukrainian Mathematical Journal Aims and scope

Abstract

We consider a criterion and sufficient conditions for the existence of a solution of the equation

$$Z_t x = \frac{{t^{n - 1} x}}{{\left( {n - 1} \right)!}} + \int\limits_0^t {a\left( {t - s} \right)AZ_s xds} $$

in a Banach space X. We determine a resolvent of the Volterra equation by differentiating the considered solution on subsets of X. We consider the notion of "incomplete" resolvent and its properties. We also weaken the Priiss conditions on the smoothness of the kernel a in the case where A generates a C 0-semigroup and the resolvent is considered on D(A).

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. J. Prüss, “Positivity and regularity of hyperbolic Volterra equations in Banach spaces,” Math. Ann., 279, 317–344 (1987).

    Article  MATH  MathSciNet  Google Scholar 

  2. J. Prüss, Evolutionary Integral Equations in Banach Spaces, Birkhauser, Basel 1993.

    Google Scholar 

  3. N. U. Ahmed, “Generalized solutions for linear systems governed by operators beyond the Hille - Yosida type,” Publ. Math. Debrecen, 48, 45–64 (1996).

    MATH  MathSciNet  Google Scholar 

  4. G. Da Prato and M. Janelli, “Linear integro-differential equations in Banach spaces,” Sem. Mat. Univ. Padova, 62, 207–219 (1980).

    MATH  Google Scholar 

  5. P. Clement and J. A. Nohel, “Abstract linear and nonlinear Volterra equations preserving positivity,” SIAM J. Math. Anal., 10, 365–388 (1979).

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics, Ukranian Academy of Sciences, Kiev

    Yu. S. Mishura

Authors
  1. Yu. S. Mishura
    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

Mishura, Y.S. Existence of solutions of abstract volterra equations in a banach space and its subsets. Ukr Math J 52, 741–753 (2000). https://doi.org/10.1007/BF02487286

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02487286

Keywords

Search

Navigation