×

Continuity of iteration and approximation of iterative roots. (English) Zbl 1219.39008

Let \(I = [a,b] \subset \mathbb{R}\). We define \(C(I)\) to be the set of continuous functions \(f: I \to I\). It is proved that the operator \(T_{n}: C(I) \to C(I)\), where \(T_{n}f = f^{n}, f^{0}(x) = x, f^{k}(x) = f(f^{k-1}(x))\) for \(x \in I, f \in C(I)\) and \(k = 1, 2,\dots,n\), is continuous with respect to the supremum metric.
Another result of this paper is the following. Let \(H(I)\) denote the set of all strictly increasing homeomorphisms of \(I\) onto \(I\) such that \(a < f(x) < x\) for \(x \in (a,b)\). If \(f, F_{m} \in H(I)\), \(F = f^{n}\) and \(F = \lim_{m \to \infty} F_{m}\), then there exists \(f_{m} \in H(I)\) such that \(F_{m} = f_{m}^{n}\) and \(\lim_{m \to \infty} f_{m} = f\) (in \(C(I)\)). The main proofs of this part of the paper are modeled after ones in the book of M. Kuczma, B. Choczewski and R. Ger [Iterative functional equations. Cambridge etc.: Cambridge University Press (1990; Zbl 0703.39005)].

MSC:

39B12 Iteration theory, iterative and composite equations
37E05 Dynamical systems involving maps of the interval
26A18 Iteration of real functions in one variable

Citations:

Zbl 0703.39005
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Essay towards the calculus of functions, II, Philos. Trans., 179-256 (1816)
[2] Bödewat, U. T., Zur iteration reeller funktionen, Math. Z., 49, 497-516 (1944) · Zbl 0028.35103
[3] Fort, M. K., The embedding of homeomorphisms in flows, Proc. Amer. Math. Soc., 6, 960-967 (1955) · Zbl 0066.41306
[4] Isaacs, R., Iterates of fractional order, Canad. J. Math., 2, 409-416 (1950) · Zbl 0039.11605
[5] Kuczma, M., On the functional equation \(\varphi^n(x) = g(x)\), Ann. Polon. Math., 11, 161-175 (1961) · Zbl 0102.11102
[6] Baron, K.; Jarczyk, W., Recent results on functional equations in a single variable, perspectives and open problems, Aequationes Math., 61, 1-48 (2001) · Zbl 0972.39011
[7] Targonski, G., Progress of iteration theory since 1981, Aequationes Math., 50, 50-72 (1995) · Zbl 0860.39029
[8] Kuczma, M., (Functional Equations in a Single Variable. Functional Equations in a Single Variable, Monograph in Mathematics, vol. 46 (1968), PWN: PWN Warsaw) · Zbl 0196.16403
[9] Kuczma, M.; Choczewski, B.; Ger, R., (Iterative Functional Equations. Iterative Functional Equations, Encycl. Math. Appl., vol. 32 (1990), Cambridge University Press: Cambridge University Press Cambridge) · Zbl 0703.39005
[10] Targonski, G., Topics in Iteration Theory (1981), Vandenhoeck and Ruprecht: Vandenhoeck and Ruprecht Göttingen · Zbl 0454.39003
[11] Blokh, A.; Coven, E.; Nitecki, Z., Roots of continuous piecewise monotone maps of an interval, Acta Math. Univ. Comenian. (NS), 60, 3-10 (1991) · Zbl 0736.58026
[12] Zhang, Jingzhong; Yang, Lu, Discussion on iterative roots of piecewise monotone functions, Acta Math. Sinica, 26, 398-412 (1983), (in Chinese) · Zbl 0529.39006
[13] Zhang, Weinian, PM functions, their characteristic intervals and iterative roots, Ann. Polon. Math., 65, 119-128 (1997) · Zbl 0873.39009
[14] Bogatyi, S., On the nonexistence of iterative roots, Topology Appl., 76, 97-123 (1997)
[15] Xu, Bing; Zhang, Weinian, Construction of continuous solutions and stability for the polynomial-like iterative equation, J. Math. Anal. Appl., 325, 1160-1170 (2007) · Zbl 1111.39020
[16] Zhang, Weinian, A generic property of globally smooth iterative roots, Sci. China Ser. A, 38, 3, 267-272 (1995) · Zbl 0838.39005
[17] Ciepliński, K., On the embedability of a homeomorphism of the circle in disjoint iteration group, Publ. Math. Debrecen, 55, 363-383 (1999) · Zbl 0935.39010
[18] Zdun, M. C., On iterative roots of homeomorphisms of the circle, Bull. Pol. Acad. Sci. Math., 48, 203-213 (2000) · Zbl 0996.39016
[19] Nikodem, K.; Zhang, Weinian, On a multivalued iterative equation, Publ. Math. Debrecen, 64, 427-435 (2004) · Zbl 1071.39024
[20] Powierża, T., On functions with weak iterative roots, Aequationes Math., 63, 103-109 (2002) · Zbl 1020.39010
[21] Iannella, N.; Kindermann, L., Finding iterative roots with a spiking neural network, Inform. Process. Lett., 95, 545-551 (2005) · Zbl 1184.68404
[22] L. Kindermann, Computing iterative roots with neural networks, in: Proceedings of the Fifth Conference on Neural Information Processing, vol. 2, 1998, pp. 713-715.; L. Kindermann, Computing iterative roots with neural networks, in: Proceedings of the Fifth Conference on Neural Information Processing, vol. 2, 1998, pp. 713-715.
[23] Kobza, J., Iterative functional equation \(x(x(t)) = f(t)\) with \(f(t)\) piecewise linear, J. Comput. Appl. Math., 115, 331-347 (2000) · Zbl 0945.65143
[24] Zhang, Wanxiong; Zhang, Weinian, Computing iterative roots of polygonal functions, J. Comput. Appl. Math., 205, 497-508 (2007) · Zbl 1120.65132
[25] Ulam, S. M., Problems in Modern Mathematics (1964), John Wiley & Sons Inc.: John Wiley & Sons Inc. New York · Zbl 0137.24201
[26] Hyers, D. H., On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA, 27, 222-224 (1941) · Zbl 0061.26403
[27] Agarwal, R.; Xu, Bing; Zhang, Weinian, Stability of functional equations in single variable, J. Math. Anal. Appl., 288, 852-869 (2003) · Zbl 1053.39042
[28] Forti, G. L., Hyers-Ulam stability of functional equations in several variables, Aequationes Math., 50, 143-190 (1995) · Zbl 0836.39007
[29] Ger, R.; S˘emrl, P., The stability of the exponential equation, Proc. Amer. Math. Soc., 124, 779-787 (1996) · Zbl 0846.39013
[30] Rassias, T. M., On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72, 297-300 (1978) · Zbl 0398.47040
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.