Solvability of a quadratic Hammerstein–Stieltjes integral equation on an unbounded interval

The paper presents some results on the existence of solutions vanishing at infinity of a quadratic Hammerstein integral equation of Stieltjes type. We investigate solutions of the mentioned integral equation which are defined, continuous and bounded on the real half-axis [Formula: see text]. The main tools used in our investigations are the technique of measures of noncompactness and a fixed point theorem of Darbo type. Moreover, we introduce the concept of the variation of a function on an unbounded interval and the integral of Stieltjes type defined on such an interval. Our result will be illustrated by an appropriate example.