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Recent developments in the translation equation and its stability. (English) Zbl 1387.39014

Brzdęk, Janusz (ed.) et al., Developments in functional equations and related topics. Selected papers based on the presentations at the 16th international conference on functional equations and inequalities, ICFEI, Będlewo, Poland, May 17–23, 2015. Cham: Springer (ISBN 978-3-319-61731-2/hbk; 978-3-319-61732-9/ebook). Springer Optimization and Its Applications 124, 215-229 (2017).
Summary: The aim of this chapter is to present some of the recent results concerning the theory of the translation equation and its stability.
For the entire collection see [Zbl 1381.39001].

MSC:

39B12 Iteration theory, iterative and composite equations
39B82 Stability, separation, extension, and related topics for functional equations
26A18 Iteration of real functions in one variable
37E10 Dynamical systems involving maps of the circle
37C15 Topological and differentiable equivalence, conjugacy, moduli, classification of dynamical systems
37E05 Dynamical systems involving maps of the interval
39B22 Functional equations for real functions
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[1] Chudziak, J.: Approximate dynamical systems on interval. Appl. Math. Lett. 25, 532-537 (2012) · Zbl 1244.39019 · doi:10.1016/j.aml.2011.09.052
[2] Ciepliński, K.: On the embeddability of a homeomorphism of the unit circle in disjoint iteration groups. Publ. Math. Debrecen 55, 363-383 (1999) · Zbl 0935.39010
[3] Ciepliński, K., Zdun, M.C.: On a system of Schr’́oder equations on a circle. Int. J. Bifurcation Chaos Appl. Sci. Eng. 13, 1883-1888 (2003) · Zbl 1062.39023 · doi:10.1142/S0218127403007709
[4] Cornfeld, I.P., Fomin, S.V., Sinai, Ya.G.: Ergodic Theory. Springer, New York (1982) · Zbl 0493.28007
[5] Hyers, D.H.: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. USA 27, 222-224 (1941) · JFM 67.0424.01 · doi:10.1073/pnas.27.4.222
[6] Jarczyk, W.: Recent results on iterative roots. In: Jarczyk, W., Fournier-Prunaret, D., Cabral, J.M.G. (eds.) ECIT 2012, 19th European Conference on Iteration Theory. ESAIM Proceedings of Surveys, vol. 46, pp. 47-62. EDP Science, Les Ulis (2014) · Zbl 1335.39001
[7] Krassowska, D., Zdun, M.C.: Embeddability of homeomorphisms of the circle in set-valued iteration groups. J. Math. Anal. Appl. 433, 1647-1658 (2016) · Zbl 1323.37026 · doi:10.1016/j.jmaa.2015.08.052
[8] Li, L., Zhang, W.M.: Conjugacy between piecewise monotonic functions and their iterative roots. Sci. China Math. 59, 367-378 (2016) · Zbl 1338.39032 · doi:10.1007/s11425-015-5065-6
[9] Liu, L., Zhang, W.: Non-monotonic iterative roots extended from characteristic intervals. J. Math. Anal. Appl. 378, 359-373 (2011) · Zbl 1216.39028 · doi:10.1016/j.jmaa.2011.01.037
[10] Mach, A., Moszner, Z.: Unstable (stable) system of stable (unstable) functional equations. Ann. Univ. Paedagog. Crac. Stud. Math. 9, 43-47 (2010) · Zbl 1298.39028
[11] Moszner, Z.: The translation equation and its application. Demonstratio Math. 6, 309-327 (1973) · Zbl 0282.39009
[12] Moszner, Z.: Sur la définition de Hyers de la stabilité de l’équation fonctionnelle. Opuscula Math. 3, 47-57 (1987) · Zbl 0654.39006
[13] Moszner, Z.: General theory of the translation equation. Aequationes Math. 50, 17-37 (1995) · Zbl 0876.39007 · doi:10.1007/BF01831111
[14] Moszner, Z.: On the stability of functional equations. Aequationes Math. 77, 33-88 (2009) · Zbl 1207.39044 · doi:10.1007/s00010-008-2945-7
[15] Moszner, Z.: On the inverse stability of functional equations. In: Brzdęk, J., Chmieliński, J., Ciepliński, K., Ger, R., Páles, Z., Zdun, M.C. (eds.) Recent Developments in Functional Equations and Inequalities. Banach Center Publications, vol. 99,, pp. 111-121. Polish Academy of Sciences, Institute of Mathematics, Warsaw (2013)
[16] Moszner, Z.: Stability has many names. Aequationes Math. 90, 983-999 (2016) · Zbl 1351.39016 · doi:10.1007/s00010-016-0429-8
[17] Moszner, Z.: On the normal stability of functional equations. Ann. Math. Silesianae 30, 111-128 (2016) · Zbl 1369.39030
[18] Moszner, Z., Przebieracz, B.: Is the dynamical system stable? Aequationes Math. 89, 279-296 (2015) · Zbl 1319.39017 · doi:10.1007/s00010-014-0330-2
[19] Przebieracz, B.: On the stability of the translation equation and dynamical systems. Nonlinear Anal. 75, 1980-1988 (2012) · Zbl 1242.39044 · doi:10.1016/j.na.2011.09.050
[20] Przebieracz, B.: A characterization of the approximate solutions of the translation equation. J. Differ. Equ. Appl. 21, 1058-1067 (2015) · Zbl 1343.39041 · doi:10.1080/10236198.2015.1065824
[21] Ulam, S.M.: A Collection of Mathematical Problems. Interscience Publishers, New York (1960) · Zbl 0086.24101
[22] Ulam, S.M.: Problems in Modern Mathematics. Science Editions Wiley, New York (1964) · Zbl 0137.24201
[23] Walters, P.: An Introduction to Ergodic Theory. Springer, New York (1982) · Zbl 0475.28009 · doi:10.1007/978-1-4612-5775-2
[24] Zdun, M.C.: On embedding of homeomorphism of the circle in continuous flow. In: Liedl, R., Reich, L., Targonski, G. (eds.) Iteration Theory and Its Functional Equations (Lochau, 1984), vol. 1163, pp. 218-231. Lecture Notes in Mathematics. Springer, Berlin (1985) · Zbl 0616.54037 · doi:10.1007/BFb0076436
[25] Zdun, M.C.: The embedding problem in iteration theory. In: Jarczyk, W., Fournier-Prunaret, D., Cabral, J.M.G. (eds.) ECIT 2012, 19th European Conference on Iteration Theory. ESAIM Proceedings Surveys, vol. 46, pp. 86-97. EDP Science, Les Ulis (2014) · Zbl 1335.37020
[26] Zdun, M.C., Solarz, P.: Recent results on iteration theory: iteration groups and semigroups in the real case. Aequationes Math. 87, 201-245 (2014) · Zbl 1295.39015 · doi:10.1007/s00010-013-0186-x
[27] Zhang, W.: PM functions, their characteristic intervals and iterative roots. Ann. Polon. Math. 65, 119-128 (1997) · Zbl 0873.39009
[28] Zhang, J.Z., Yang, L.: Iterative roots of a piecewise monotone continuous self-mapping. (Chinese) Acta Math. Sinica 26, 398-412 (1983) · Zbl 0529.39006
[29] Zhang, J.Z., Yang, L., Zhang, W.: Some advances on functional equations. Adv. Math. (China) 24, 385-405 (1995) · Zbl 0862.39009
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