Advertisement

Ukrainian Mathematical Journal

, Volume 52, Issue 5, pp 741–753 | Cite as

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

  • Yu. S. Mishura
Article

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).

Keywords

BANACH Space Laplace Transformation VOLTERRA Equation Exponential Estimate Closed Linear Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    J. Prüss, “Positivity and regularity of hyperbolic Volterra equations in Banach spaces,” Math. Ann., 279, 317–344 (1987).zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    J. Prüss, Evolutionary Integral Equations in Banach Spaces, Birkhauser, Basel 1993.Google Scholar
  3. 3.
    N. U. Ahmed, “Generalized solutions for linear systems governed by operators beyond the Hille - Yosida type,” Publ. Math. Debrecen, 48, 45–64 (1996).zbMATHMathSciNetGoogle Scholar
  4. 4.
    G. Da Prato and M. Janelli, “Linear integro-differential equations in Banach spaces,” Sem. Mat. Univ. Padova, 62, 207–219 (1980).zbMATHGoogle Scholar
  5. 5.
    P. Clement and J. A. Nohel, “Abstract linear and nonlinear Volterra equations preserving positivity,” SIAM J. Math. Anal., 10, 365–388 (1979).zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Kluwer Academic/Plenum Publishers 2000

Authors and Affiliations

  • Yu. S. Mishura
    • 1
  1. 1.Institute of MathematicsUkranian Academy of SciencesKiev

Personalised recommendations