An iterative method for solving one class of non-linear integral equations with Nemytskii operator on the positive semi-axis

Abstract

A class of non-linear integral equations with monotone Nemytskii operator on the positive semi-axis is considered. This class of integral equations appears in many areas of modern natural science. In particular, such equations, with various restrictions on non-linearity and the kernel, arise in the dynamic theory of $p$-adic strings for the scalar field of tachyons, in the kinetic theory of gases and plasma within the framework of the usual and modified non-linear Bhatnagar-Gross-Crook models for the Boltzmann kinetic equation. Equations of similar nature appear also in the non-linear radiative transfer theory in inhomogeneous media and in the mathematical theory of the spread of epidemic diseases within the framework of the modified Diekmann-Kaper model. A constructive theorem for the existence of a bounded positive and continuous solution is proved. As a result, we get a uniform estimate of the difference between the previous and subsequent iterations, the corresponding successive approximations converge uniformly to a bounded continuous solution to the equation. The asymptotic behaviour of the constructed solution at infinity is studied. In particular, it is proved that the limit of this solution at infinity exists and is positive, and is uniquely determined from a certain characteristic equation. It is also proved that the difference between the limit and the solution is a summable function on the positive semi-axis. Using some geometric estimates for convex and concave functions, and employing the theorem on integral asymptotics obtained here, we prove the uniqueness of the solution in a certain subclass of non-negative non-trivial continuous and bounded functions. The results obtained are also applied to the study of a special class of non-linear Urysohn type integral equations on the positive semi-axis. In particular, the existence of a positive bounded solution to this class of equations is verified, and some qualitative properties of the constructed solution are studied. We also give specific applied examples of the corresponding kernels and non-linearities to illustrate the importance of the results obtained.