ON THE SOLVABILITY OF A SYSTEM OF MULTIDIMENSIONAL INTEGRAL EQUATIONS WITH CONCAVE NONLINEARITIES

The work is devoted to the study of questions of existence and uniqueness of a continuous bounded and positive solution to one system of nonlinear multidimensional integral equations. The scalar analogue 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 and bounded solution to the system under study. It is possible to prove 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 wide subclass of continuous bounded and coordinately nonnegative vector functions. In the case when the integral of the matrix kernel has a unit spectral radius, it is proved that in a certain subclass of continuous bounded and coordinate-wise non-negative vector functions, this system has only a trivial solution, which is an eigenvector of the kernel integral matrix.