Solvability of a Volterra–Stieltjes integral equation in the class of functions having limits at infinity

Abstract

The paper is devoted to the study of the solvability of a nonlinear Volterra–Stieltjes integral equation in the class of real functions defined, bounded and continuous on the real half-axis $\mathbb{R}_+$ and having finite limits at infinity. The considered class of integral equations contains, as special cases, a few types of nonlinear integral equations. In particular, that class contains the Volterra–Hammerstein integral equation and the Volterra–Wiener–Hopf integral equation, among others. The basic tools applied in our study is the classical Schauder fixed point principle and a suitable criterion for relative compactness in the Banach space of real functions defined, bounded and continuous on $\mathbb{R}_+$. Moreover, we will utilize some facts and results from the theory of functions of bounded variation.

Keywords

• space of continuous and bounded functions
• variation of function
• function of bounded variation
• riemann–stieltjes integral
• criterion of relative compactness
• integral equation
• schauder fixed point principle