Solvability of a nonlinear Volterra–Stieltjes integral equation in the class of bounded and continuous functions of two variables

Abstract

The aim of this paper is to investigate the solvability of a nonlinear Volterra–Stieltjes integral equation in two variables in the space \(BC(\mathbb {R_+} \times \mathbb {R_+},\mathbb {R})\). We will consider this equation in the space of real continuous and bounded functions on the set \(\mathbb {R_+} \times \mathbb {R_+}\). Applying the fixed point theorem of Darbo type and the technique of measure of noncompactness we will show that solutions of the mentioned equation exist. We will also give the characterization of these solutions. At the end of our considerations we will introduce some special cases of the studied equation and we will include an example illustrating our main result.