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

M. Federson, R. Bianconi

Address. M. Federson, Instituto de Ciencias Matematicas e de Computacao, Universidade de Sao Paulo, CP 688, Sao Carlos, SP 13560-970, BRAZIL

R. Bianconi, Instituto de Matematica e Estatistica, Universidade de Sao Paulo, CP 66281, Sao Paulo, SP 05315-970, BRAZIL

E-mail: federson@icmc.sc.usp.br

             bianconi@ime.usp.br

Abstract. In 1990, H\"onig 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 H\"onig'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.

AMSclassification. 45A05, 26A39

Keywords. Linear integral equations, Kurzweil-Henstock integrals