On non-trivial solvability of one system of non-linear integral equations on the real axis

A system of singular integral equations with monotonic and convex non-linearity on the entire real line is considered. System of this form have applications in many areas of natural science. In particular, such systems arise in the theory of $p$-adic open-closed strings, in the mathematical theory of spatial-temporal epidemic spread within the framework of the well known Diekmann–Kaper model, in the kinetic theory of gases, in the radiative transfer theory. An existence theorem for a non-trivial and bounded solution is proved. The asymptotic behaviour of the constructed solution at $\pm\infty$ is also studied. Specific examples of non-linearities and kernel functions having an applied character are given.