One-parameter Family of Positive Solutions for a Nonlinear Integral Equation Arising in Physical Kinetics

Abstract

The paper is devoted to the question of solvability of a Urysohn type nonlinear integral equation. This equation has an application in the kinetic theory of gases and can be derived from Boltzmann model equation. We prove an existence theorem of one-parameter family of positive solutions in the space of functions possessing linear growth at infinity. Moreover, for each member of this family we find an exact asymptotic formula at infinity. We obtain two-sided estimates for solution, as well as describe an iterative method for construction of solution.We conclude the paper by giving examples of functions that describe nonlinearity and satisfy the conditions of the main theorem.