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Composition iterates, Cauchy, translation, and Sincov inclusions. (English) Zbl 1457.39013

The authors – by improving and extending some ideas of Gottlob Frege going back to 1874 – study some nice properties of the composition iterates of a function. They also consider translation functions and Sincov inclusions. The paper presents nice results in general topology of functions and it could inspire further research in the subject.

MSC:

39B12 Iteration theory, iterative and composite equations
39B52 Functional equations for functions with more general domains and/or ranges
39B62 Functional inequalities, including subadditivity, convexity, etc.
20N02 Sets with a single binary operation (groupoids)
26E25 Set-valued functions
54E25 Semimetric spaces
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References:

[1] J. Aczél, Lectures on Functional Equations and Their Applications, Academic Press, New York, 1966. · Zbl 0139.09301
[2] M. Alimohammady, S. Jafari, S. P. Moshokoa and M. K. Kalleji, A note on properties of hypermetric spaces, J. Hyperstructures, 3 (2014), 89-100. · Zbl 1329.54028
[3] P. Augustová and L. Klapka, Atlas as solutions of Sincov’s inequality, arXiv: 1612.00355v1 [math.DS] 1 Dec 2016, 7 pp.
[4] J. A. Baker, Solution of problem E 2607, Amer. Math. Monthly, 84 (1977), 824-825.
[5] G. Birkhoff, Lattice Theory, Amer. Math. Soc. Colloq. Publ., 25, Providence, Rhode Island, 1967. · Zbl 0153.02501
[6] H. Brandt, Über eine Verallgemeinerung der Gruppenbregriffes, Math. Ann., 96 (1926), 360-366. · JFM 52.0110.09
[7] M. J. Campión, R. G. Catalán, E. Induráin and G. Ochoa, Reinterpreting a fuzzy subset by means of a Sincov’s functional equation, J. Intelligent Fuzzy Systems, 27 (2014), 367-375. · Zbl 1309.03021
[8] M. J. Campión, E. Induráin, G. Ochoa and O. Valero, Functional equations related to weightable quasi-metrics, Hacet. J. Math. Stat., 44 (2015), 775-787. · Zbl 1333.39018
[9] M. Cantor, Funktionalgleichungen mit drei von einander unabhängigen Veränderlichen, Zeitschrift Mat. Physik., 41 (1896), 161-163. · JFM 27.0312.01
[10] E. Castillo-Ron and R. Ruiz-Cobo, Functional Equations in Science and Engineering, Marcel Decker, New York, 1992. · Zbl 0801.39004
[11] R. Croisot, Une interprétation des relations d’équivalence dans un ensemble, C. R. Acad. Sci. Paris, 226 (1948), 616-617. · Zbl 0030.29401
[12] B. A. Davey and H. A. Priestley, Introduction to Lattices and Order, Cambridge University Press, Cambridge, 2002. · Zbl 1002.06001
[13] W. Fechner, Richard’s inequality, Cauchy-Schwarz’s inequality and approximate solutions of Sincov’s equation, Proc. Amer. Math. Soc., 147 (2019), 3955-3960. · Zbl 1428.26038
[14] W. Fechner, Sincov’s inequalities on topological spaces, Publ. Math. Debrecen, 96 (2020), 63-76. · Zbl 1463.39059
[15] G. Frege, Rechnungsmethoden, die sich auf eine Erweiterung des Grössenbegriffes gründen, Dissertation for the Venia docendi, Verlag Friedrich Frommann, Jena, 1874.
[16] G. Frege, Methods of calculation based on an extension of the concept of quantity, In: G. Frege, Collected Papers of on Mathematics, Logic, and Philosophy, Basil Blackwell, Oxford, 1984, 56-92.
[17] Z. Gajda, Invariant means and representations of semigroups in the theory of functional equations, Prace Naukowe Uniwersytetu Śląskiego w Katowicach, 1273 (1992), 1-81. · Zbl 0925.39005
[18] T. Glavosits, Generated preorders and equivalences, Acta Acad. Paed. Agriensis, Sect. Math., 29 (2002), 95-103. · Zbl 1012.08002
[19] T. Glavosits and Á. Száz, Constructions and extensions of free and controlled additive relations, In: Th.M. Rassias (Ed.), Handbook of Functional Equations: Functional Inequalities, Springer Optim. Appl. 95 (2014), 161-208. · Zbl 1322.39009
[20] D. Gronau, Gottlob Frege, a pioneer in iteration theory, In: L. Reich, J. Smítal and Gy. Targonski (Eds.), Proceedings of the European Conference on Iteration Theory, ECIT94, Gracer Math. Ber. 334 (1997), 105-119. · Zbl 0918.01010
[21] D. Gronau, Gottlob Fregees Beiträge zur Iteratiostheorie und zur Theorie der Functionalgleichungen, In: G. Gabriel (Ed.), Gottlob Frege - Werk und Wirkung, Mentis Verlag, Paderborn, 200, 151-169.
[22] D. Gronau, A remark on Sincov’s functional equation, Notices South African Math. Soc. Esaim: Proceeding and Surveys, 31 (2000), 1-8.
[23] D. Gronau, Translation equation and Sincov’s equation - A historical remark, Esaim: Proceeding and Surveys 46 (2014), 43-46. · Zbl 1330.01049
[24] J. Mala and Á. Száz, Modifications of relators, Acta Math. Hungar., 77 (1997), 69-81. · Zbl 0902.08001
[25] Z. Moszner, Solution générale de l’équation F(x, y)F(y, z)= F(x, z) pour x ≤ y ≤ z, C. R. Acad. Sci. Paris, 261 (1965), 28. · Zbl 0132.11501
[26] Z. Moszner, L’équation de translation et l1équation de Sincov généralisée, Rocznik Nauk.-Dydakt. Prace Mat., 16 (1999), 53-71. · Zbl 1160.39317
[27] Z. Moszner, On the stability of functional equations, Aequationes Math., 77 (2009), 33-88. · Zbl 1207.39044
[28] K. Nikodem, Additive selections of additive set-valued functions, Zb. Rad. Prirod.-Mat. Fak., 18 (1988), 143-148. · Zbl 0686.46008
[29] K. Nikodem and D. Popa, On single-valuedness of set-valued maps satisfying linear inclusions, Banach J. Math. Anal., 3 (2009), 44-51. · Zbl 1163.26353
[30] G. Pataki and Á. Száz, A unified treatment of well-chainedness and connectedness properties, Acta Math. Acad. Paedagog. Nyházi. (N.S.), 19 (2003), 101-165. · Zbl 1049.54023
[31] J. Pepis, Sur une famille d’ensembles plans et les solutions de l’équation fonctionnelle F(x, z)= F(x, y) · F(y, z) pour 0 ≤ x ≤ y ≤ z. Applicationá la théorie générale des intéréts, Ann. Soc. Polon. Math., 17 (1937), 113.
[32] W. J. Pervin, Quasi-uniformization of topological spaces, Math. Ann., 147 (1962), 316-317. · Zbl 0101.40501
[33] B. Piatek, On the Sincov functional equation, Demonstratio Math., 38 (2005), 875-881. · Zbl 1092.39026
[34] P. K. Sahoo, Stability of a Sincov type functional equation, J. Inf. Math. Sci., 1 (2009), 81-90. · Zbl 1205.39030
[35] P. K. Sahoo, On a Sincov type functional equation, In: Th.M. Rassias and J. Brzdek (Eds.), Functional Equations in Mathematical Analysis, Springer Optimizations and Its Applications, 52, Chapter 43, Springer, New York, 2012, 697-7008.
[36] D. M. Sincov, Über eine Funktionalgleichung, Arch. Math. Phys., 6 (1904), 216-217. · JFM 34.0421.03
[37] A. Smajdor, Iterations of multi-valued functions, Prace Naukowe Universytetu Ślaskiego w Katowicach 759, Universitet Ślaski, Katowice 1985. · Zbl 0561.39006
[38] W. Smajdor, Set-valued version of Sincov’s functional equation, Demonstration Math., 39 (2006), 101-105. · Zbl 1095.39021
[39] Á. Száz, Galois type connections and closure operations on preordered sets, Acta Math. Univ.Comen., 78 (2009), 1-21. · Zbl 1199.06005
[40] Á. Száz, Corelations are more powerful tools than relations, In: Th.M. Rassias (Ed.), Applications of Nonlinear Analysis, Springer Optimization and Its Applications, 134 (2018), 711-779. · Zbl 1504.08002
[41] A. Weil, Sur les espacesá structure uniforme at sur la topologie générale, Actual. Sci. Ind., 551 Herman and Cie, Paris, 1937. · JFM 63.0569.04
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