On the constructive solvability of a nonlinear Volterra integral equation on the entire real line

A nonlinear integral equation with a Hammerstein–Volterra operator on the entire real line is considered. A constructive existence theorem for a bounded and continuous solution is established. Moreover, the uniform convergence of successive approximations to the solution is proved, with the error decreasing at a geometric rate. The integral asymptotics of the constructed solution are then investigated. Additionally, the uniqueness of the solution is demonstrated within a specific subclass of bounded and continuous functions. Finally, specific examples of equations and nonlinearities satisfying all the conditions of the theorems are provided.