ON NONTRIVIAL SOLVABILITY OF ONE CLASS OF NONLINEAR INTEGRAL EQUATIONS WITH CONSERVATIVE KERNEL ON THE POSITIVE SEMI-AXIS

Abstract

The work is devoted to a special class of nonlinear integral equations on the positive semi-axis with conservative kernel that corresponds to a nonlinear operator, for which the property of complete continuity in the space of bounded functions fails. In different special cases this class of equations has applications in particular branches of mathematical physics. In particular, this kind of equations can be met in the radiative transfer theory, kinetic theory of gases, kinetic theory of plasma and in the $p$-adic open-closed string theory. Using a combination of special iterations with the monotonic operator theory methods, that work in defined conical segments it is possible to prove a constructive existence theorem of nonnegative nontrivial bounded solution that has finite limit at infinity. The asymptotics of the constructed solution will also be studied. It is also given an example of nonlinear equation, for which the uniqueness of the solution in the space of bounded functions fails. At the end of the paper will consider some classes of equations both applied and pure theoretical character.