In the paper the question of solvability of an Urysohn type nonlinear integral equation arising in kinetic theory of gases has been studied. We prove the existence of a positive and bounded solution and also suggest an approach for the construction of a solution. We also show that there is a qualitative difference between solutions in the linear and nonlinear cases. In the nonlinear case the solution is a posi- tive and bounded function, while the corresponding linear equation has an alternating solution, which possesses linear growth at infinity.