Geodesic
On the fixed points of monotonic operators in the critical case
Izvestiya. Mathematics , Tome 70 (2006) no. 5, pp. 931-947

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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. 
@article{IM2_2006_70_5_a3,
     author = {N. B. Engibaryan},
     title = {On the fixed points of monotonic operators in the critical case},
     journal = {Izvestiya. Mathematics },
     pages = {931--947},
     publisher = {mathdoc},
     volume = {70},
     number = {5},
     year = {2006},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2006_70_5_a3/}
}
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N. B. Engibaryan. On the fixed points of monotonic operators in the critical case. Izvestiya. Mathematics , Tome 70 (2006) no. 5, pp. 931-947. http://geodesic.mathdoc.fr/item/IM2_2006_70_5_a3/