Continuous solutions for a Volterra-Stieltjes quadratic integral equation (Q819569)

The author considers a quadratic Volterra-Stieltjes integral equation in the space \(BC(\mathbb{R}_+)\) consisting of real functions defined, continuous and bounded on \(\mathbb{R}_+= [0,+\infty)\) with supremum norm. It is shown that the mentioned integral equation has a solution in \(BC(\mathbb{R}_+)\) under some conditions. The main tool used in the proof is the technique of measures of noncompactness. The results of the paper seems to be not valid since the author uses a function being a kind of the modulus of continuity of a set as the measure of noncompactness in the space \(BC(\mathbb{R}_+)\). It is well-known that such a function is not a measure of noncompactness in \(BC(\mathbb{R}_+)\).