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On coincidence points of mappings in generalized metric spaces

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Let X be a space with ∞-metric ρ (a metric with possibly infinite value) and Y a space with ∞-distance d satisfying the identity axiom. We consider the problem of coincidence point for mappings F,G:X→Y, i.e. the problem of existence of a solution for the equation F(x)=G(x). We provide conditions of the existence of coincidence points in terms of a covering set for the mapping F and a Lipschitz set for the mapping G in the space X×Y. An α-covering set (α>0) of the mapping F is a set of (x,y) such that ∃u∈X F(u)=y, ρ(x,u)≤α^(-1) d(F(x),y), ρ(x,u)<∞, and a β - Lipschitz set (β≥0) for the mapping G is a set of (x,y) such that ∀u∈X G(u)=y⇒d(y,G(x))≤βρ(u,x). The new results are compared with the known theorems about coincidence points.

Keywords

coincidence point of two mappings; metric; distance; covering mapping

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DOI

10.20310/2686-9667-2020-25-129-18-24

UDC

517.988.6, 515.124.2

Pages

18-24

References

[1] A.V. Arutyunov, “Covering mappings in metric spaces and fixed points”, Doklady Mathematics, 76:2 (2007), 665–668. [2] A.V. Arutyunov, A.V. Greshnov, “Theory of (q1,q2)-quasimetric spaces and coincidence points”, Doklady Mathematics, 94:1 (2016), 434–437. [3] E.S. Zhukovskiy, “On Coincidence Points of Multivalued Vector Mappings of Metric Spaces”, Mathematical Notes, 100:3 (2016), 363–379. [4] E.S. Zhukovskiy, “On Coincidence Points for Vector Mappings”, Russian Mathematics, 60:10 (2016), 10–22. [5] E.S. Zhukovskiy, E.A. Pluzhnikova, “Covering mappings in a product of metric spaces and boundary value problems for differential equations unsolved for the derivative”, Differential Equations, 49:4 (2013), 420–436. [6] W. Merchela, “About Arutyunov theorem of coincidence point for two mapping in metric spaces”, Tambov University Reports. Series: Natural and Technical Sciences, 23:121 (2018), 65–73 (In Russian). [7] S. Benarab, E.S. Zhukovskii, W. Merchela, “Theorems on perturbations of covering mappings in spaces with a distance and in spaces with a binary relation”, Trudy instituta matematiki i mekhaniki UrO RAN, 25:4 (2019), 52–63 (In Russian). [8] E.S. Zhukovskiy, “The fixed points of contractions of f-quasimetric spaces”, Siberian Mathematical Journal, 59:6 (2018), 1063–1072

Received

2019-12-23

Section of issue

Scientific articles

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