迭代函数方程的同胚解

Abstract

Abstract:

Most known results on the solutions of iterative functional equations are about their continuity and smoothness. However, concerning the study of conjugation and linearization in functional equations and dynamical systems, the property of homeomorphism play an important role in these theories. Although partial results were obtained, a full description is still open. In this paper, we first consider homeomorphic solutions to a general iterative functional equation by the methods of construction and approximation. Then, the obtained results are used to study the solutions of polynomial-like iterative functional equations.

Key words: Iterative equation, Homeomorphic solution, Locally linear function, Uniform convergence, Homogeneous function

CLC Number:

O175.23

Liu Jinghua, Li Lin. Homeomorphic Solutions of Iterative Functional Equations[J].Acta mathematica scientia,Series A, 2024, 44(2): 313-325.

Trendmd

Figures/Tables 10

References 35

[1]	Baron K, Jarczyk W. Recent results on functional equations in a single variable perspectives and open problems. Aequationes Math, 2001, 61: 1-48
[2]	Bessis D, Marmi S, Turchetti G. On the singularities of divergent majorant series arising from normal form theory. Rend Mat Appl, 1989, 9: 645-659
[3]	Block L, Coven M. Topological conjugacy and transitivity for a class of piecewise monotone maps of the interval. Trans Amer Math Soc, 1987, 300: 297-306
[4]	Chen Jingmin, Zhang Weinian. Leading coefficient problem for polynomial-like iterative equations. J Math Anal Appl, 2009, 349: 413-419
[5]	Draga S, Morawiec J. Reducing the polynomial-like iteative equations order and a generalized Zoltán Boros' Problem. Aequationes Math, 2016, 90: 935-950
[6]	Draga S, Morawiec J. Mean of iterates. Aequationes Math, 2019, 93: 21-35
[7]	Fontich E. Transversal homoclinic points of a class of conservative diffeomorphisms. J Differential Equations, 1990, 87: 1-27
[8]	Greenfield S J, Nussbaum R D. Dynamics of a quadratic map in two complex variables. J Differential Equations, 2001, 169: 57-141
[9]	Hu J, Sullivan D P. Topological conjugacy of circle diffeomorphisms. Ergodic Theory Dyn Syst, 1997, 17(1): 173-186 doi: 10.1017/S0143385797061002
[10]	Jarczyk W. On an equation of linear iteration. Aequationes Math, 1996, 51: 303-310
[11]	Kuczma M, Choczewski B, Ger R. Iterative Functional Equations. Encyclopedia Math Appl, Vol 32. Cambridge:Cambridge University Press, 1990
[12]	Lanford O. A computer-assisted proof of the Feigenbaum conjectures. Bull Amer Math Soc (NS), 1982, 6: 427-434
[13]	Li L, Song W. Global solutions for leading coefficient problem of a general iterative equation. Results Math, 2015, 68(1/2): 247-260 doi: 10.1007/s00025-014-0432-0
[14]	Li L, Zhang W M. Continuously decreasing solutions for polynomial-like iterative equations. Sci China Ser A, 2013, 56: 1051-1058
[15]	Li M R, Zhao H Y. Strongly convex solutions of polynomial-like iterative equation. J Math Anal Appl, 2021, 495(2): 124786 doi: 10.1016/j.jmaa.2020.124786
[16]	Liu J H. Analytical invariant curves for a second order difference equation modeled from macroeconomics under the Brjuno condition. J Difference Equ Appl, 2017, 3: 648-656
[17]	Liu J H, Yu Z H, Zhang W N. Invariant curves for a second order difference equation modeled from macroeconomics. J Difference Equ Appl, 2015, 9: 757-773
[18]	Matkowski J, Weinian Z. Method of characteristic for function equations in polynomial form. Acta Math Sinica, 1997, 13: 421-432
[19]	McCarthy P. The general exact bijective continuous solution of Feigenbaum's functional equation. Comm Math Phys, 1983, 91: 431-443
[20]	Nabeya S. On the functional equation $ f(p + qx + rf(x)) = a + bx + cf(x) $. Aequationes Math, 1974, 11: 199-211
[21]	Parry W. Symbolic dynamics and transformations of the unit interval. Trans Amer Math Soc, 1966, 122: 368-378
[22]	Song W, Li L. Continuously decreasing solutions for a general iterative equation. Acta Mathematica Scientia, 2018, 38B: 177-186
[23]	Tabor J, Tabor J. On a linear iterative equation. Results Math, 1995, 27: 412-421
[24]	Xu B, Zhang W N. Construction of continuous solutions and stability for the polynomial-like iterative equation. J Math Anal Appl, 2007, 325: 1160-1170
[25]	Yang D L, Zhang W N. Characteristic solutions of polynomial-like iterative equations. Aequationes Math, 2004, 67: 80-105
[26]	Yang L, Zhang J Z, The second type of Feigenbaum's functional equation. Sci China Ser A, 1986, 29: 1252-1262
[27]	Zhang J Z, Yang L, Zhang W N. Some advances on functional equations. Adv Math (China), 1995, 24: 385-405
[28]	Zhang W N. Discussion on the iterated equation $ \Sigma_{i=1}^{n}\lambda_{i}f^{i}(x)=F(x) $. Chinese Sci Bull, 1987, 32: 1444-1451
[29]	Zhang W N. Discussion on the differentiable solutions of the iterated equation $ \Sigma_{i=1}^{n}\lambda_{i}f^{i}(x)=F(x) $. Nonlinear Anal, 1990, 15: 387-398
[30]	Zhang W N, Baker J. Continuous solutions of a polynomial-like iterative equation with variable coefficients. Ann Polon Math, 2000, 73: 29-36
[31]	Zhang W N, Nikodem K, Xu B. Convex solutions of polynomial-like iterative equations. J Math Anal Appl, 2006, 315: 29-40
[32]	Zhang W M, Xu B, Zhang W N. Global solutions for leading coefficient problem of polynomial-like iterative equations. Results Math, 2013, 63: 79-93
[33]	Zhang W M, Zhang W N. On continuous solutions of $ n $-th order polynomial-like iterative equations. Publ Math Debrecen, 2010, 76: 117-134
[34]	Zhang W M, Zhang W N. $ C^{1} $ linearization for planar contractions. J Funct Anal, 2011, 260: 2043-2063 doi: 10.1016/j.jfa.2010.12.029
[35]	Zhang W M, Zhang W N, Jarczyk W. Sharp regularity of linearization for $ C^{1,1} $ hyperbolic diffeomorphisms. Math Ann, 2014, 358: 69-113

Homeomorphic Solutions of Iterative Functional Equations

RichHTML

PDF (PC)

Abstract

Cite this article

share this article

Figures/Tables 10

References 35

Related Articles 9

Metrics

Comments

Recommended 10

[1]	Zhiheng Yu,Xiaobing Gong. Polynomial Solutions of the Polynomial-Like Iterative Equation [J]. Acta mathematica scientia,Series A, 2019, 39(6): 1352-1364.
[2]	YANG Yu-Bo, ZHU Peng, YIN Yun-Hui. A Optimal Uniformly Convergent Discontinuous Galerkin Finite Element Method for Singularly Perturbed Problem [J]. Acta mathematica scientia,Series A, 2014, 34(3): 716-726.
[3]	YANG Zheng, JI Yuan-Yuan, MA He-Ping. Uniform Convergence of the Fourier Spectral Method for the Zakharov Equations [J]. Acta mathematica scientia,Series A, 2013, 33(3): 409-423.
[4]	YUN Dong-Fang, HUANG Shu-Xiang. On a Class of Singular Equations with Nonlinear Terms Depending on the Gradient [J]. Acta mathematica scientia,Series A, 2012, 32(5): 861-878.
[5]	ZHONG Ji-Yu, LI Xiao-Pei. On Set-valued Solutions of an Iterative Equation [J]. Acta mathematica scientia,Series A, 2011, 31(4): 970-977.
[6]	TU Tian-Liang, MO Jiong. Interpolatory Approximation to Harmonic Function with Boundary Data [J]. Acta mathematica scientia,Series A, 2010, 30(2): 397-404.
[7]	Peng Zhigang. The Properties of Several Classes of Analytic Functions with Missing Coefficients [J]. Acta mathematica scientia,Series A, 2008, 28(4): 661-669.
[8]	BANG Zhi-Gang, SU Feng-. Extreme Points and Support Points of a Family of Analytic Func tions [J]. Acta mathematica scientia,Series A, 2005, 25(3): 345-348.
[9]	Zhu Zhongyi, Wei Bocheng. Strong Uniform Convergence Rates for Nonparametric Estimations of Conditional Functional and its Derivatives [J]. Acta mathematica scientia,Series A, 1998, 18(4): 394-401.