Linear Integral Equations of Volterra Concerning Henstock Integrals

Abstract

We establish conditions for the existence of solutions of the linear integral equation of Volterra \begin{equation} x\left( t\right) +^{\ast }\int\nolimits_{[ a,t] }\alpha ( s) x ( s )\, ds=f ( t ) ,\quad t\in [ a,b ] ,\tag{$V_{\ast}$} \end{equation} where the functions are Banach space-valued and $^{\ast }\int $ denotes either the Bochner-Lebesgue or the Henstock integral. In some cases it is possible to calculate the solution of $( V)_{\ast }$ explicitly. We give several examples.