Solvability of a certain system of singular integral equations with convex nonlinearity on the positive half-line

We study a system of nonlinear singular integral equations with a sum-difference kernel on the positive half-line. In various representations, the system arises in many branches of mathematical physics and applied mathematics. In particular, a system of equations with a kernel representing a Gaussian distribution and with power nonlinearity arises in the dynamic theory of $p$-adic open-closed strings, and in the case when the nonlinearity has a certain exponential structure, such a system occurs in mathematical biology, namely in the theory of the spatio-temporal distribution of the epidemic.
The constructive theorems of the existence of non-negative non-trivial continuous and bounded solutions are proved. The questions of uniqueness and asymptotic behavior of the constructed solutions at infinity are investigated. At the end, specific applied examples of these equations are given that satisfy all the conditions of the proved theorems.