Existence of solutions of abstract Volterra equations in a Banach space and its subsets (Q2722123)

The author considers the following equation in a Banach space \(X\): NEWLINE\[NEWLINEZ_{t}x=t^{n-1}x/(n-1)!+\int_{0}^{t}a(t-s)AZ_{s}x\, ds,NEWLINE\]NEWLINE \(x\in X\). A criterion and sufficient conditions for the existence of a solution to the equation are proposed. The resolvent of the Volterra equation by differentiating the considered solution on subsets of \(X\) is obtained. NEWLINENEWLINENEWLINEConditions of \textit{J. Prüss} [Evolutionary integral equations and applications. Monographs in Mathematics. 87. Basel: Birkhäuser Verlag (1993; Zbl 0784.45006)] on the smoothness of the kernel \(a(t-s)\) in the case where \(A\) generates a \(C_0\)-semigroup and the resolvent is considered on \(D(A)\), are weakened.