On solution of a system of Hammerstein–Nemitskii type nonlinear integral equations on whole axis

We study a system of Hammerstein–Nemitskii type nonlinear integral equations on whole axis. The system has direct application in the theory of Ricker`s nonlinear system for travelling waves.The combination of special iterative methods with the theory of primitive matrices and properties of convolution operations allow us to prove existence of a positive solution (by components) in the space of integrable functions possesing zero limit at infinity.