Abstract
A class of nonlinear multidimensional Urysohn-type integral equations in an \(n\) -dimensional Euclidean space is studied in the critical case. For various representations of the Urysohn kernel, these equations arise in the theory of \(p\) -adic strings and in the mathematical theory of epidemic spread. A constructive existence theorem for a nontrivial, positive, continuous, and bounded solution is established. It is shown that the associated successive approximations converge uniformly to the solution with a rate of a decreasing geometric progression. Moreover, in a certain class of bounded functions, a uniqueness theorem for the obtained solution is established. At the end of the paper, concrete examples of such equations are presented to illustrate the significance of the results.
