On the Solvability of a System of Multidimensional IntegralEquations with Concave Nonlinearities

The paper studies the existence and uniqueness of a continuous bounded positive solution of a system of nonlinear multidimensional integral equations. The scalar analog of the indicated system of integral equations, with different representations of the corresponding matrix kernel and nonlinearities, has important applied significance in a number of areas of physics and biology. This article proposes a special iterative approach for constructing a positive continuous bounded solution to the system under study. It is shown that the corresponding iterations uniformly converge to a continuous solution of the specified system. Using some a priori estimates for strictly concave functions, we also prove the uniqueness of the solution in a fairly broad subclass of continuous bounded and coordinatewise nonnegative vector functions. For the case in which the integral of the matrix kernel has unit spectral radius, we prove that, in a certain subclass of continuous bounded and coordinatewise nonnegative vector functions, this system has only a trivial solution that is an eigenvector of the integral kernel matrix.