Zero Preservation for a Family of Multivalued Functionals, and Applications to the Theory of Fixed Points and Coincidenc
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Zero Preservation for a Family of Multivalued Functionals, and Applications to the Theory of Fixed Points and Coincidences T. N. Fomenkoa,* and Yu. N. Zakharyanb,** Presented by Academician of the RAS S.V. Matveev May 23, 2020 Received May 23, 2020; revised May 23, 2020; accepted June 2, 2020
Abstract—A theorem on the zero existence preservation for a parametric family of multivalued (α, β)-search functionals on an open subset of a metric space is proved. Several corollaries on the existence preservation for preimages of a closed subspace, for coincidence points, and for common fixed points under the action of a parametric family (a number of families) of mappings are obtained. The notion of a Zamfirescu-type pair of mappings is introduced, and a coincidence theorem for such pairs of mappings is obtained. In addition, a theorem on the coincidence existence preservation for a parametric family of such pairs of mappings is obtained. The obtained results imply several well-known theorems. Keywords: parametric family of functionals, multivalued mapping, fixed point, coincidence point, Zamfirescu mapping, Zamfirescu-type pair of mappings, preservation of coincidence existence, parametric family of mappings DOI: 10.1134/S1064562420040225
Given a parametric family of multivalued (α,β) search functionals on an open subset of a metric space, we prove a theorem on the preservation of the property of this family to have zeros under variations in the numerical parameter. The proof is based on one of the versions of the cascade search principle for zeros of functionals proposed in [1, 2]. From this main result, for a given open subset, we derive several corollaries on existence preservation for preimages of a closed subspace of a metric image space under the action of a parametric family of multivalued mappings and on existence preservation for coincidence points and common fixed points of a finite collection of parametric families of multivalued mappings. For families of multivalued mappings of ordered sets, similar problems concerning the preservation of the existence of fixed and coincidence points were considered in [3]. Additionally, we introduce a new concept of a Zamfirescu-type pair of multivalued mappings. A theorem on the existence of coincidence points for such a pair of mappings is obtained that generalizes the
results of [4], and a theorem on the coincidence existence preservation for a parametric family of Zamfirescu-type pairs of multivalued mappings under variations in the numerical parameter is derived. As a simple special case, the above-mentioned results imply the Frigon–Granas fixed point theorem for a contractive family of multivalued mappings [5, 6]. First, we give the necessary notation. Let ( X , d ) and (Y , ρ) be metric spaces. Let C ( X ) denote the set of nonempty closed subsets of X, and CB( X ) be the set of nonempty closed bounded subsets of X. The distance between any nonempty subsets A, B of X is defined as
d ( A, B ) := inf{d(a, b)| a ∈ A, b ∈ B},
the distance from a point a to a su
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