Existence and uniqueness theorems for one system of integral equations with two nonlinearities

We consider a system of integral equations on the positive semiaxis with two monotone nonlinearities. With various particular representations of matrix kernels and nonlinearities, this system arises in many branches of mathematical physics. A constructive existence theorem for a non-negative, non-trivial and bounded solution is proved. We also study the asymptotic behavior of the solution at infinity. Under additional restrictions on the nonlinearities and matrix kernels, a uniqueness theorem for a solution, in a certain class of bounded vector functions, is proved. At the end, specific examples of matrix kernels and nonlinearities are given.