Constructive study of the solvability of one class of nonlinear integral equations with a symmetric kernel

A class of nonlinear integral equations of Hammerstein type with a symmetric and substochastic kernel is considered. This class of equations is encountered in many branches of physics and mathematical biology. In particular, equations of this nature arise in the dynamic theory of a $p$-adic string, in the kinetic theory of gases, and in various model problems of the mathematical theory of the spread of epidemic diseases. A constructive theorem on the existence of a nontrivial bounded nonnegative continuous and monotonically nondecreasing solution is proved. In the class of nontrivial nonnegative and bounded functions, the uniqueness of the solution is also proved. The results obtained are used to study a nonlinear integral equation on the entire line with an almost difference kernel. At the end of the paper, particular examples of the kernel and nonlinearity are given that are of applied nature in the above theories.