On some systems of nonlinear integral equations on the whole axis with monotonous Hammerstein–Volterra type operators

The work is devoted to studying and solving some systems of nonlinear integral equations with monotonous Hammerstein–Volterra integral operators. In specific cases of matrix kernels and nonlinearities the specified systems have applications in various fields of mathematical physics and mathematical biology. Firstly, a quasilinear system of integral equations on the whole axis with monotonous nonlinearity will be investigated, and a constructive theorem of existence of a one-parameter family of componentwise nonnegative (nontrivial) bounded solutions will be proved. Then, the asymptotic behaviour of the constructed solutions will be studied at $-\infty$. Then, using the obtained results, a system of integral equations with two nonlinearities with different characteristics will be investigated. Under certain limitations on the first nonlinearity we will prove the existence of componentwise nonnegative and bounded solution for such systems. In addition, the limit of the constructed solution at $-\infty$ will be calculated, and the asymptotics of the difference between the limit and the solution will be established. At the end of this paper specific examples of matrix kernels and nonlinearities will be given for the illustration of the obtained results.