Positive solutions to a nonlinear Abel-type integral equation on the wholeline

We consider the existence of positive solutions to the nonlinear integral e quation u(x) = integral (x)(-infinity) (x - s)(alpha -1)g(u(s))ds, (x is an element of R, alpha greater than or equal to 1), 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) > 0 for all x > -infinity. We also provide a condition which ensures that all the nontrivial solutions experience the blow-up behaviour. (C) 2001 Elsevi er Science Ltd. All rights reserved.