Positive fixed points of Hammerstein integral operators with degenerate kernel

Yu. Kh. Eshkabilov   ; Sh. D. Nodirov

Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 166 (2024) no. 3, pp. 437-449

Résumé

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.

Détail

Keywords: fixed point, Hammerstein integral operator, Cayley tree, Gibbs measure, translation-invariant Gibbs measure.

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