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On the fixed points of monotonic operators in the critical case. (English. Russian original) Zbl 1131.45006

Izv. Math. 70, No. 5, 931-947 (2006); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 70, No. 5, 79-96 (2006).
Summary: 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 \(\|\varphi x\|\leq\|x\|+\gamma\), \(\gamma>0\), 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.

MSC:

45P05 Integral operators
45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
45G05 Singular nonlinear integral equations
47H07 Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces
47H10 Fixed-point theorems
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