Geodesic
Positive fixed points of Hammerstein integral operators with degenerate kernel
Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 166 (2024) no. 3, pp. 437-449

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Positive fixed points of the Hammerstein integral operators with a degenerate kernel in the space of continuous functions $C[0,1]$ were explored. The problem of determining the number of positive fixed points of the Hammerstein integral operator was reduced to analyzing the positive roots of polynomials with real coefficients. A model on a Cayley tree with nearest-neighbor interactions and with the set $[0,1]$ of spin values was considered. It was proved that a unique translation-invariant Gibbs measure exists for this model.
Keywords: fixed point, Hammerstein integral operator, Cayley tree, Gibbs measure, translation-invariant Gibbs measure. 
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     author = {Yu. Kh. Eshkabilov and Sh. D. Nodirov},
     title = {Positive fixed points of {Hammerstein} integral operators with degenerate kernel},
     journal = {U\v{c}\"enye zapiski Kazanskogo universiteta. Seri\^a Fiziko-matemati\v{c}eskie nauki},
     pages = {437--449},
     publisher = {mathdoc},
     volume = {166},
     number = {3},
     year = {2024},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/UZKU_2024_166_3_a11/}
}
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Yu. Kh. Eshkabilov; Sh. D. Nodirov. Positive fixed points of Hammerstein integral operators with degenerate kernel. Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 166 (2024) no. 3, pp. 437-449. http://geodesic.mathdoc.fr/item/UZKU_2024_166_3_a11/