Uniqueness for branching diffusion process with absorbing boundary

With suitable assumptions, uniqueness of nontrivial nonnegative solution is established for the following nonlinear integral equation phi(x) = integral(D) E(x, y)G(y,phi(y)) dy. This result is applied to branching diffusion processes with absorbing boundary. In particular, a question existing in the study by Pazy and Rabinowitz [A nonlinear integral equation with applications to neutron transport theory. Arch Ration Mech Anal. 1969;32:226-246] is solved.