Abstract:
Nadler's theorem on a fixed point of a multivalued mapping is extended to spaces with a vector-valued metric. A vector-valued metric is understood as a mapping with the properties of a usual metric and values in a linear normed ordered space. We prove an analog of Nadler's theorem and apply it to a system of integral inclusions in a space of summable functions. Then we study a boundary value problem with multivalued conditions for systems of functional differential equations by means of reduction to a system of integral inclusions. Conditions for the existence of solutions are obtained and estimates of the solutions are given. The existence conditions do not contain the convexity requirement for the values of the multivalued function generating a Nemytskii operator.
Keywords:
space with a vector-valued metric, contracting multivalued mapping, fixed point, integral inclusion.
Citation:
E. S. Zhukovskiy, E. A. Panasenko, “On fixed points of multivalued mappings in spaces with a vector-valued metric”, Trudy Inst. Mat. i Mekh. UrO RAN, 24, no. 1, 2018, 93–105; Proc. Steklov Inst. Math. (Suppl.), 305, suppl. 1 (2019), S191–S203
\Bibitem{ZhuPan18}
\by E.~S.~Zhukovskiy, E.~A.~Panasenko
\paper On fixed points of multivalued mappings in spaces with a vector-valued metric
\serial Trudy Inst. Mat. i Mekh. UrO RAN
\yr 2018
\vol 24
\issue 1
\pages 93--105
\mathnet{http://mi.mathnet.ru/timm1499}
\crossref{https://doi.org/10.21538/0134-4889-2018-24-1-93-105}
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\transl
\jour Proc. Steklov Inst. Math. (Suppl.)
\yr 2019
\vol 305
\issue , suppl. 1
\pages S191--S203
\crossref{https://doi.org/10.1134/S0081543819040199}
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Linking options:
https://www.mathnet.ru/eng/timm1499
https://www.mathnet.ru/eng/timm/v24/i1/p93
This publication is cited in the following 11 articles:
V. Obukhovskii, T. Ul'vacheva, “On a Class of Condensing Multivalued Maps”, Lobachevskii J Math, 45:1 (2024), 491
T. N. Fomenko, “Zeros of Conic Functions, Fixed Points, and Coincidences”, Dokl. Math., 2024
T. N. Fomenko, “Zeros of conic functions, fixed points and coincidences”, Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ, 517:1 (2024), 74
E. A. Panasenko, “On Operator Inclusions in Spaces with Vector-Valued Metrics”, Proc. Steklov Inst. Math. (Suppl.), 323, suppl. 1 (2023), S222–S242
Evgeny Zhukovskiy, Elena Panasenko, “Extension of the Kantorovich Theorem to Equations in Vector Metric Spaces: Applications to Functional Differential Equations”, Mathematics, 12:1 (2023), 64
E. S. Zhukovskiy, “A Note on Generalized Contraction Theorems”, Math. Notes, 111:2 (2022), 211–216
Aram V. Arutyunov, Evgeny S. Zhukovskiy, Sergey E. Zhukovskiy, Zukhra T. Zhukovskaya, “Kantorovich's Fixed Point Theorem and Coincidence Point Theorems for Mappings in Vector Metric Spaces”, Set-Valued Var. Anal, 30:2 (2022), 397
E. S. Zhukovskii, “O probleme suschestvovaniya nepodvizhnoi tochki obobschenno szhimayuschego mnogoznachnogo otobrazheniya”, Vestnik rossiiskikh universitetov. Matematika, 26:136 (2021), 372–381
T. V. Zhukovskaya, E. A. Pluzhnikova, “Mnozhestvo regulyarnosti mnogoznachnogo otobrazheniya v prostranstve s vektornoznachnoi metrikoi”, Vestnik rossiiskikh universitetov. Matematika, 24:125 (2019), 39–46
E. S. Zhukovskiy, “The fixed points of contractions of f-quasimetric spaces”, Siberian Math. J., 59:6 (2018), 1063–1072
E. A. Pluzhnikova, T. V. Zhukovskaya, Yu. A. Moiseev, “O mnozhestvakh metricheskoi regulyarnosti otobrazhenii v prostranstvakh s vektornoznachnoi metrikoi”, Vestnik Tambovskogo universiteta. Seriya: estestvennye i tekhnicheskie nauki, 23:123 (2018), 547–554
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