On the fixed points of monotonic operators in the critical case

We consider the problem of constructing positive fixed points $x$ of monotonic operators $\varphi$ acting on a cone $K$ in a Banach space $E$. We assume that $\|\varphi x\|\le\|x\|+\gamma$, $\gamma>0$, for all $x\in K$. In the case when $\varphi$ has a so-called non-trivial dissipation functional we construct a solution in an extension of $E$, which is a Banach space or a Fréchet 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.