On solvability of a nonlinear problem in theory of income distribution

We consider a nonlinear integro-differential equation with a Hammerstein type noncompact operator, arising in the theory of income distribution. We prove the existence of a positive solution of the nonlinear problem in Sobolev space $W_1^1(\mathbb R^+)$. We list some examples arising in applications. For one modeling problem a uniqueness theorem is proved. At the end of the paper the results of numerical calculations are given.