An existence theorem for nonlinear Volterra integral equation with deviating argument (Q579574)

The paper deals with nonlinear Volterra integral equations with deviating argument \[ x(t)=h(t)+\int^{t}_{0}K(t,s,x(H(s)))ds,\quad t\geq 0, \] in \(C_ g\) spaces: \(C_ g(R_+,R^ n)=\{x: R_ 0\to R^ n\), continuous and \(| x(t)| /g(t)\) bounded on \(R_+\}\). It is assumed that g is a continuous positive function on \(R_+\). Under various conditions on the data it is shown that the equation under discussion has at least one solution in a convenient \(C_ g\) space (using Schauder fixed point theorem). The classical case of Volterra equation is covered by the result.