Properties of stationary solutions of a generalized Tjon–Wu equation

A generalized version of the Tjon–Wu equation is considered. Solutions of this equation are functions with values in the space of probability measures on [0,∞). We prove that the stationary solution μ∗ of the equation has the following property: Either μ∗ is supported at one point or suppμ∗=[0,∞). Moreover we show that in the second case the distribution function of μ∗ is continuous. Some open questions are discussed.