On summable solutions to two-dimensional Volterra integral equations with monotone nonlinearity on a quarter of the plane (Q2104317)

The authors study the equation \[ B(x,y)=\int_x^\infty\int_y^\infty V(x'-x,y'-y) \bigl (G(B(x',y'))+ \omega(x',y',B(x',y'))\bigr)\, dx'\, dy', \] where \((x,y)\in \mathbb R^+\times \mathbb R^+\) and show that there is a positive integrable solution. Here the function \(G\) is increasing, upward convex, \(G(0)=0\) and such that there is a number \(\eta>0\) with \(G(\eta)=\eta\). For each \((x,y)\in \mathbb R^+\times \mathbb R^+\) the function \(u\mapsto \omega(x,y,u)\) is increasing and \(\omega(x,y,0)=0\) and the function \(V\) is positive, bounded and satisfies \(\int_0^\infty\int_0^\infty V(x,y)\,dx\, dy=1\). In addition there are a number of technical conditions connecting the functions \(G\), \(\omega\) and \(V\). From the arguments used for the proof one gets certain exponential estimates on the solution as well. Some examples are given.