On the Constructive Solvability of a System of Hammerstein–Volterra Nonlinear Integral Equationson the Positive Semiaxis

A system of nonlinear integral equations with the Hammerstein–Volterra matrix operator is investigated. The specified system of equations, in addition to purely mathematical interest, is of particular interest in various fields of natural science. In particular, such equations are encountered in hydroaerodynamics, in population genetics models, and in the theory of radiative heat transfer. A constructive theorem on the existence of a nonnegative bounded and continuous solution to the specified system of equations is proved. With one additional constraint on the nonlinearity, uniform convergence of specially selected successive approximations with the rate of decreasing geometric progression is obtained. The asymptotic behavior of the constructed solution at infinity is also investigated. Moreover, a theorem on the uniqueness of a solution in the class of bounded vector functions whose coordinates are nonnegative is proved. At the end of the work, specific examples of the specified systems are given that satisfy all the conditions of the proved statements.