Abstract:
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.
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