Abstract
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.
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Zakharyan, Y.N., Fomenko, T.N. Preservation of the Existence of Zeros in a Family of Set-Valued Functionals and Some Consequences. Math Notes 108, 802–813 (2020). https://doi.org/10.1134/S0001434620110231
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DOI: https://doi.org/10.1134/S0001434620110231