Solvability of a System of Nonlinear Integral Equations on the Entire Line

TL;DR

This paper investigates the existence, uniqueness, and asymptotic behavior of solutions to a system of nonlinear integral equations with applications in physics and biology, introducing an effective iterative solution method.

Contribution

It provides a constructive existence theorem, studies asymptotics, and proposes an iterative process for solving nonlinear integral systems with applications in science.

Findings

Existence of solutions in the space of continuous and bounded vector functions.

Uniqueness of solutions in a certain class of bounded vector functions.

Non-existence of zero solutions under specific conditions.

Abstract

A system of singular integral equations with monotone and concave nonlinearity in the subcritical case is investigated. The specified system and its scalar analog have direct applications in various areas of physics and biology. In particular, scalar and vector equations of this nature are encountered in the dynamic theory of p-adic strings, in the theory of radiative transfer, in the kinetic theory of gases, and in the mathematical theory of the spread of epidemic diseases. A constructive theorem of existence in the space of continuous and bounded vector functions is proved. The integral asymptotic of the constructed solution is studied. An effective iterative process for constructing an approximate solution to this system is proposed. In a certain class of bounded vector functions, the uniqueness of the solution is proved. In the class of bounded vector functions (with non-negative coordinates) with a zero limit at infinity, the absence of an identically zero solution is proved. Examples of the matrix kernel and nonlinearities are given to illustrate the importance of the obtained results. Some of the examples have applications in the above-mentioned branches of natural science.