Study of the existence and uniqueness of a bounded solution of a system of nonlinear multidimensional integral equations in the critical case

The article studies a system of nonlinear multidimensional integral equations in the first octant. This system and its scalar analogue have applications in various branches of mathematical physics. In particular, such nonlinear integral equations are encountered in physical kinetics, in the theory of radiative transfer and in the dynamical theory of $p$-adic open-stopped strings. Under certain restrictions on kernels and nonlinearities, a constructive theorem of the existence of a bounded nonnegative continuous and continuously differentiable solution to the given system is proved. Moreover, under additional conditions on kernels and nonlinearities, uniform convergence of the corresponding successive approximations to the solution with the rate of decrease of the geometric progression is established. The uniqueness of the constructed solution in a certain subclass of bounded vector functions with nonnegative coordinates is also proved. At the end of the paper, specific examples of kernels and nonlinearities that satisfy all the conditions of the formulated theorems are given.