On the Constructive Solvability of One Class of Nonlinear Integral Equations of the Hammerstein Type on the Whole Line

The work is devoted to numerical solution and study of some qualitative properties of the solution of one class of nonlinear integral equations on the whole line with a noncompact and monotone Hammerstein-type operator. 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 is proved on the existence of a continuous positive and bounded solution having the same finite limit at $$ \pm \infty .$$ 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 to the equation under study. With additional restriction on the kernel, it is also proved that the difference between the solution and its limit value at $$ \pm \infty $$ is an integrable function on the entire number line. Uniqueness of the solution in the class of nonnegative nontrivial continuous and bounded functions is obtained from the 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.