Geodesic
Karisti inequality and $\alpha$-contractive mappings
Funkcionalʹnyj analiz i ego priloženiâ, Tome 53 (2019) no. 3, pp. 84-88

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The article considers a new Caristi-like inequality and proves some development of the Caristi theorem on fixed points of mappings of complete metric spaces (both in the single-valued and multi-valued case). Based on the obtained theorem, we study mappings of complete metric spaces that are contractive with respect to a certain $\alpha$ function of 2 vector arguments $\alpha$-contractive mappings). This function may not be a metric or even a continuous function. Proved theorems are generalizations of the Banach principle of contraction maps of and the Nadler theorem.
Keywords: fixed point, multivalued mapping, metric space, contraction mappings. 
@article{FAA_2019_53_3_a6,
     author = {B. D. Gel'man},
     title = {Karisti inequality and $\alpha$-contractive mappings},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {84--88},
     publisher = {mathdoc},
     volume = {53},
     number = {3},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2019_53_3_a6/}
}
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B. D. Gel'man. Karisti inequality and $\alpha$-contractive mappings. Funkcionalʹnyj analiz i ego priloženiâ, Tome 53 (2019) no. 3, pp. 84-88. http://geodesic.mathdoc.fr/item/FAA_2019_53_3_a6/