On a class of nonlinear integral equations of the Hammerstein–Volterra type on a semiaxis

In this note, we study a class of nonlinear integral equations with a monotone Hammerstein-Volterra type operator in the critical case. This class of equations occurs in the kinetic theory of gases in the framework of the study of the nonlinear kinetic integro-differential model Boltzmann equation. The combination of methods for constructing invariant cone segments for a nonlinear monotone operator with the methods of the theory of functions of a real variable makes it possible, with the help of specially chosen successive approximations, to construct a positive summable and bounded solution on a non-negative semiaxis for the above class of equations. With an additional constraint on nonlinearity, it is also possible to prove the uniqueness of the solution in a certain class of positive and summable functions on the non-negative semiaxis. At the end, illustrative examples of nonlinearity and the kernel are given, which are of both theoretical and applied interest.