On a Volterra Stieltjes integral equation (Q916046)

Assume that \[ \int_{K}| k(s,t)| \{dV_ s(\alpha_ 1)dV_ t(\beta_ 2)+dV_ s(\alpha_ 2)dV_ t(\beta_ 1)+dV_ s(\alpha_ 2)dV_ t(\beta_ 2)\}<1, \] where k is continuous on \(K=[a_ 1,b_ 1]\times [a_ 2,b_ 2]\), \(\alpha\) and \(\beta\) are of bounded variation, right-continuous having only isolated discontinuities, and \(V_ s\), \(V_ t\), \(V_ x\) being the total variations on \([a_ 1,s]\), \([a_ 2,t]\) and [a,x] resp. \((a=(a_ 1,a_ 2))\), then \[ u(x)=g(x)+\int_{[a,x]}k(s,t)u(s,t)d\alpha (s)d\beta (t), \] where g is of bounded variation, has a unique solution u, which can be obtained by successive approximation starting with g. An example is given involving other Bessel functions than \(I_ 0\).