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


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).


BANACH Space Laplace Transformation VOLTERRA Equation Exponential Estimate Closed Linear Operator 
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Copyright information

© Kluwer Academic/Plenum Publishers 2000

Authors and Affiliations

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

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