On the constructive solvability of one class nonlinear integral equations of the Hammerstein type on the whole line

The work is devoted to numerical solution and to the study of some qualitative properties of the solution of one class of nonlinear integral equations on the whole line with non-compact and monotone operator of Hammerstein type. This class of equations has applications in various areas of physics and epidemiology. In particular, under certain representations of the corresponding kernel and nonlinearity, such equations arise in the theory of $p$-adic strings, in the kinetic theory of gases, and in the mathematical theory of propagation epidemic diseases within different models. With certain restrictions on kernel and on the nonlinearity of the equation, a constructive theorem on the existence of a continuous positive and bounded solution having the same finite limit on $\pm\infty.$ In addition, we obtain an estimate for the difference of the corresponding neighboring successive approximations, from which it follows that these approximations in terms of the speed of a geometric progression uniformly converge to a continuous and bounded solution of the equation under study. With additional restriction on the kernel, it is also proved that the difference between the solution and its limit value on $\pm\infty$ is an integrable function on the entire number line. Uniqueness of the solution in the class of non-negative non-trivial continuous and bounded functions is obtained from previously known results of the authors of this paper. At the end of the work, numerical calculations are given for some model examples of the kernel and nonlinearity.