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

  • Yu. S. Mishura Київ. нац. ун-т iм. Т. Шевченка

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).
Published
25.05.2000
How to Cite
MishuraY. S. “Existence of Solutions of Abstract Volterra Equations in a Banach Space and Its Subsets”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 52, no. 5, May 2000, pp. 648-57, https://umj.imath.kiev.ua/index.php/umj/article/view/4459.
Section
Research articles