Sufficient conditions for the solvability of the Urysohn integral equation on a half-line (Q735841)

Given a continuous function \(K:\mathbb R^+\times\mathbb R^+\times\mathbb R\to\mathbb R^+\), the author gives conditions under which the Urysohn equation \[ f(x)=\int^\infty_0 K(x,t,f(x))\,dt \quad (x>0) \] has a bounded solution. The main hypothesis is the a priori estimate \[ \int^\infty_0 K(x,t,\eta)\,dt\leq \eta \quad (x>0) \] for some \(\eta>0\).