On the Solvability of One Infinite System of Integral Equations with Power Nonlinearity on the Semi-Axis

Abstract

An infinite system of integral equations with power nonlinearity on the positive half-line is considered. A number of particular cases of this system arise in many branches of mathematical physics. In particular, systems of this nature are encountered in the theory of radiative transfer in spectral lines, in the dynamic theory of \(p\)-adic open-closed strings, in the mathematical theory of the spread of epidemic diseases, and in econometrics. The existence of a nonnegative (in coordinates) nontrivial and bounded solution is proved. Under an additional constraint on the matrix kernel, we also study the asymptotic behavior at infinity. In the case of strong symmetry (symmetry both in coordinates and in indices) of the matrix kernel, we also prove a uniqueness theorem for a solution in a certain class of infinite and bounded vector functions. At the end, concrete examples of an infinite matrix kernel are given that are of practical interest in the above applications.