Linear Volterra-Stieltjes integral equations in the sense of the Kurzweil-Henstock integral

Federson, M. ; Bianconi, R.

Archivum mathematicum, Tome 37 (2001) no. 4, pp. 307-328.

Résumé

In 1990, Hönig proved that the linear Volterra integral equation \[ x\left( t\right) -\,(K)\int \nolimits _{\left[ a,t\right] }\alpha \left( t,s\right) x\left( s\right)\,ds=f\left( t\right)\,,\qquad t\in \left[ a,b\right]\,, \] where the functions are Banach space-valued and $f$ is a Kurzweil integrable function defined on a compact interval $\left[ a,b\right] $ of the real line $\mathbb R$, admits one and only one solution in the space of the Kurzweil integrable functions with resolvent given by the Neumann series. In the present paper, we extend Hönig’s result to the linear Volterra-Stieltjes integral equation \[ x\left( t\right) - (K)\int \nolimits _{\left[ a,t\right] }\alpha \left( t,s\right) x\left( s\right) dg\left( s\right) =f\left( t\right) ,\qquad t\in \left[ a,b\right]\,, \] in a real-valued context.

Détail

Classification : 26A39, 45A05
Keywords: linear integral equations; Kurzweil-Henstock integrals