Positive solutions to a nonlinear abel-type integral equation on the whole line

We consider the existence of positive solutions to the nonlinear integral equation where g is a continuous, nondecreasing function such that g(0) = 0. We show that the equation always has nontrivial solutions and we give a necessary and sufficient condition for the existence of solutions u such that u(x) > − ∞. We also provide a condition which ensures that all the nontrivial solutions experience the blow-up behaviour.