On an Integral Equation with Concave Nonlinearity

Abstract

A nonlinear integral equation on the semi-axis with a special substochastic kernel is studied. Such equations are encountered in the kinetic theory of gases when studying the nonlinear integro-differential Boltzmann equation within the framework of the nonlinear modified Bhatnagar-Gross-Crook model(BGC). Under certain restrictions on nonlinearity, it is possible to construct a positive continuous and bounded solution to this equation. Moreover, the uniqueness of the solution in the class of upper bounded on half-line functions having a positive infimum. It is also proved that the corresponding successive approximations converge uniformly at a rate of some geometric progression to the solution of the indicated equation. Under one additional condition, the asymptotic behavior of the solution at infinity is studied. At the end of the work, specific examples of these equations are given for which all the conditions of the proven facts are automatically met.

Keywords

• concavity
• iterations
• monotonicity
• convergence
• asymptotics