Abstract <p> The paper studies the existence and uniqueness of a continuous bounded positive solutionof a system of nonlinear multidimensional integral equations. The scalar analog of the indicatedsystem of integral equations, with different representations of the corresponding matrix kernel andnonlinearities, has important applied significance in a number of areas of physics and biology. Thisarticle proposes a special iterative approach for constructing a positive continuous boundedsolution to the system under study. It is shown that the corresponding iterations uniformlyconverge to a continuous solution of the specified system. Using some a priori estimates forstrictly concave functions, we also prove the uniqueness of the solution in a fairly broad subclass ofcontinuous bounded and coordinatewise nonnegative vector functions. For the case in which theintegral of the matrix kernel has unit spectral radius, we prove that, in a certain subclass ofcontinuous bounded and coordinatewise nonnegative vector functions, this system has onlya trivial solution that is an eigenvector of the integral kernel matrix.</p>