Geodesic
Preservation of the Existence of Zeros in a Family of Set-Valued Functionals and Some Consequences
Matematičeskie zametki, Tome 108 (2020) no. 6, pp. 837-850

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A theorem on the preservation of the existence of zeros under the change of the parameter in a one-parameter family of $(\alpha,\beta)$-search functionals on an open subset of a metric space is proved. The following corollaries of this theorem are presented: on the preservation of the existence of preimages of a given closed subspace in a parametric family of multivalued mappings of metric spaces; on the preservation of the existence of coincidence points in a finite collection of two or more families of multivalued mappings of metric spaces; on the preservation of the existence of common fixed points in a collection of families of multivalued mappings to itself of a metric space. As a simple particular case, the Frigon–Granas theorem (1994) on fixed points of a contraction family of multivalued mappings is obtained.
Keywords: search functional, family of set-valued functionals, fixed point, coincidence point, contraction family of mappings. 
@article{MZM_2020_108_6_a2,
     author = {Yu. N. Zakharyan and T. N. Fomenko},
     title = {Preservation of the {Existence} of {Zeros} in a {Family} of {Set-Valued} {Functionals} and {Some} {Consequences}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {837--850},
     publisher = {mathdoc},
     volume = {108},
     number = {6},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2020_108_6_a2/}
}
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Yu. N. Zakharyan; T. N. Fomenko. Preservation of the Existence of Zeros in a Family of Set-Valued Functionals and Some Consequences. Matematičeskie zametki, Tome 108 (2020) no. 6, pp. 837-850. http://geodesic.mathdoc.fr/item/MZM_2020_108_6_a2/