Characterization of continuous additive set-valued maps “modulo K” on finite dimensional linear spaces
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Eliza Jabłońska
Abstract
Let Y be a real vector metric space and K ⊂ Y be a closed convex cone such that K ∩ (− K) = {0}. We prove that a convex compact-valued map F : ℝ → 2Y ∖ {∅} is K-continuous and K-additive if and only if there are non-empty convex compact sets A, B ⊂ Y such that 0 ∈ A − B ⊂ K and F is equal “modulo K” to the continuous set-valued map
Next, we use this result to characterize convex compact-valued maps F : ℝN → 2Y ∖ {∅}.
The research of the author was partially supported by the Faculty of Applied Mathematics AGH UST statutory tasks and dean grant within subsidy of Ministry of Science and Higher Education.
Acknowledgement
The author would like to thank to the anonymous Referee for his valuable suggestions and remarks.
Communicated by L’ubica Holá
References
[1] Aubin, J. P.—Frankowska, H.: Set-Valued Analysis, Birkhäuser, Boston–Basel–Berlin, 1990.Search in Google Scholar
[2] Bouligand, G.: Introduction à la Géométrie Infinitésimale Directe, Vuibert, Paris, 1932.Search in Google Scholar
[3] Jabłońska, E.—JabŁoński, W.: Properties of K-additive set-valued maps, Results Math. 78 (2023), Art. 221.10.1007/s00025-023-02002-5Search in Google Scholar
[4] Jabłońska, E.—Nikodem, K.: K-subadditive and K-superadditive set-valued functions bounded on “large” sets, Aequationes Math. 95 (2021), 1221–1231.10.1007/s00010-021-00850-6Search in Google Scholar
[5] JabŁońska, E.—Nikodem, K.: Further remarks on local K-boundedness of K-subadditive set-valued maps, Aequationes Math. 97 (2023), 981–993.10.1007/s00010-023-01017-1Search in Google Scholar
[6] Kakutani, S.: A generalization of Brouwer’s Fixed Point Theorem, Duke Math. J. 8 (1941), 457–459.10.1215/S0012-7094-41-00838-4Search in Google Scholar
[7] Kuczma, M.: An Introduction to the Theory of Functional Equations and Inequalities. Cauchy’s Equation and Jensen’s Inequality, PWN – Uniwersytet Śląski, Warszawa–Kraków–Katowice, 1985. 2nd edition: Birkhäuser, Basel–Boston–Berlin, 2009.10.1007/978-3-7643-8749-5Search in Google Scholar
[8] Kuratowski, K.: Topologie, Instytut Matematyczny PAN, Warszawa–Lwów, 1933.Search in Google Scholar
[9] Michael, E.: Continuous selections. I. Ann. of Math. 63 (1956), 361–382.10.2307/1969615Search in Google Scholar
[10] Nadler, S. B.: Multi-valued contraction mappings, Pacific J. Math. 30 (1969), 475–488.10.2140/pjm.1969.30.475Search in Google Scholar
[11] Nikodem, K.: K-convex and K-concave Set-valued Functions, Zeszyty Nauk. Politechniki Łódzkiej Mat. 559; Rozprawy Mat. 114, Łódź, 1989.Search in Google Scholar
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Articles in the same Issue
- On monoids of endomorphisms of a cycle graph
- The influence of separating cycles in drawings of K5 ∖ e in the join product with paths and cycles
- Completely hereditarily atomic OMLS
- New notes on the equation d(n) = d(φ(n)) and the inequality d(n) > d(φ(n))
- On the monogenity and Galois group of certain classes of polynomials
- Other fundamental systems of solutions of the differential equation (D2 – 2αD + α2 + β2)ny = 0, β ≠ 0
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- Hermite–Hadamard inequalities for Riemann–Liouville fractional integrals
- Global Mild Solutions For Hilfer Fractional Neutral Evolution Equation
- The discrete fractional Karamata theorem and its applications
- Complex Generalized Stancu-Schurer Operators
- New oscillatory criteria for third-order differential equations with mixed argument
- Cauchy problem for a loaded hyperbolic equation with the Bessel operator
- Liouville type theorem for a system of elliptic inequalities on weighted graphs without (p0)-condition
- On the rate of convergence for the q-Durrmeyer polynomials in complex domains
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