Solutions of a quadratic Volterra–Stieltjes integral equation in the class of functions converging at infinity

Abstract

The paper deals with the study of the existence of solutions of a quadratic integral equation of Volterra–Stieltjes type. We are looking for solutions in the class of real functions continuous and bounded on the real half-axis $\mathbb{R}_+$ and converging to proper limits at infinity. The quadratic integral equations considered in the paper contain, as special cases, a lot of nonlinear integral equations such as Volterra–Chandrasekhar or Volterra–Wiener–Hopf equations, for example. In our investigations we use the technique associated with measures of noncompactness and the Darbo fixed point theorem. Particularly, we utilize a measure of noncompactness related to the class of functions in which solutions of the integral equation in question are looking for.

Keywords

• space of continuous and bounded functions
• variation of function
• function of bounded variation
• riemann–stieltjes integral
• measure of noncompactness
• darbo fixed point theorem