On the fixed points of monotonic operators in the critical case

We consider the problem of constructing positive fixed points x of monotonic operators phi acting on a cone K in a Banach space E. We assume that parallel to phi x parallel to <= parallel to x parallel to + gamma, gamma > 0, for all x is an element of K. In the case when phi has a so-called non-trivial dissipation functional we construct a solution in an extension of E, which is a Banach space or a Frechet space. We consider examples in which we prove the solubility of a conservative integral equation on the half-line with a sum-difference kernel, and of a non-linear integral equation of Urysohn type in the critical case.