Volterra-Stieltjes integral operators (Q2486808)

The authors show that, under suitable growth and monotonicity assumptions on the data, the Volterra-Stieltjes integral operator \[ Vx(t) =\int^t_0v\bigl(s,x(s) \bigr)\,ds\,g(t,s)\quad (0\leq t\leq 1) \] maps the space \(C[0,1]\) into the space \(BV[0,1]\) (resp. \(BV[0,1]\cap C[0,1])\) and is both continuous and compact. This allows them to prove existence, through Schauder's fixed point theorem, of continuous monotone solutions \(x\) of the equation \(x=p+Vx\).