Existence of Solutions Converging at Infinity of a Quadratic Hammerstein–Stieltjes Integral Equation

The aim of the paper is to establish the existence of solutions of a quadratic Hammerstein–Stieltjes integral equation considered on the real half-axis R+. We seek solutions in the class of real-valued functions which are continuous and bounded on R+ and possess finite limits at infinity. The proof is based on the technique of measures of noncompactness combined with the fixed-point theorem of Darbo type. Our considerations also involve the concepts of bounded variation and the Riemann–Stieltjes integral on an unbounded interval. The obtained result extends previous investigations concerning solutions tending to zero at infinity.