December  2020, 40(12): 6967-6984. doi: 10.3934/dcds.2020214

Interval homeomorphic solutions of a functional equation of nonautonomous iterations

1. 

Department of Mathematics, College of Liberal Arts and Sciences, National University of Defense Technology, Changsha, Hunan 410073, China

2. 

School of Mathematical Sciences, Sichuan Normal University, Chengdu, Sichuan 610068, China

3. 

Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China

* Corresponding author: Weinian Zhang

Received  June 2019 Revised  January 2020 Published  May 2020

Fund Project: Zeng is supported by NSFC grant #11501394 and the Science Research Fund of Sichuan Provincial Education Department #15ZB0041. Zhang is supported by NSFC grants #11771307, #11831012 and #11821001

In this paper we study a functional equation of linear combination of nonautonomous iterations, which can be reduced from invariance of an interval homeomorphism under nonautonomous iteration. We discuss existence, uniqueness and continuous dependence for increasing Lipschitzian solutions on a compact interval, and also discuss for bounded increasing Lipschitzian solutions and unbounded ones on the whole $ \mathbb{R} $.

Citation: Xiao Tang, Yingying Zeng, Weinian Zhang. Interval homeomorphic solutions of a functional equation of nonautonomous iterations. Discrete & Continuous Dynamical Systems - A, 2020, 40 (12) : 6967-6984. doi: 10.3934/dcds.2020214
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W. Zhang, Stability of the solution of the iterated equation $\sum_{i=1}^{n}\lambda_if^i(x)=F(x)$, Acta Math. Sci. English Series, 8 (1988), 421-424.  doi: 10.1016/S0252-9602(18)30318-7.  Google Scholar

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W. Zhang, Discussion on the differentiable solutions of the iterated equation $\sum_{i=1}^n \lambda_i f^i(x)= F(x)$, Nonlinear Anal., 15 (1990), 387-398.  doi: 10.1016/0362-546X(90)90147-9.  Google Scholar

[19]

W. Zhang and J. A. Baker, Continuous solutions of a polynomial-like iterative equation with variable coefficients, Ann. Polon. Math., 73 (2000), 29-36.  doi: 10.4064/ap-73-1-29-36.  Google Scholar

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show all references

References:
[1]

R. Brück and M. Büger, Generalized iteration, Comput. Methods Funct. Theory, 3 (2003), 201-252.  doi: 10.1007/BF03321035.  Google Scholar

[2]

M. Comerford, Hyperbolic non-autonomous Julia sets, Ergod. Th. & Dynam. Syst., 26 (2006), 353-377.  doi: 10.1017/S0143385705000441.  Google Scholar

[3]

P. Cull, M. Flahive and R. Robson, Difference Equations: From Rabbits to Chaos, Springer, New York, 2005.  Google Scholar

[4]

S. Elaydi, An Introduction to Difference Equations, 3$^rd$ edition, Springer, New York, 2005.  Google Scholar

[5]

J. Fornaess and N. Sibony, Random iterations of rational functions, Ergod. Th. & Dynam. Syst., 11 (1991), 687-708.  doi: 10.1017/S0143385700006428.  Google Scholar

[6]

R. Geiselhart and F. Wirth, Solving iterative functional equations for a class of piecewise linear ${\mathcal K}_{\infty}$-functions, J. Math. Anal. Appl., 411 (2014), 652-664.  doi: 10.1016/j.jmaa.2013.10.016.  Google Scholar

[7]

W. Jarczyk, On an equation of linear iteration, Aequationes Math., 51 (1996), 303-310.  doi: 10.1007/BF01833285.  Google Scholar

[8]

M. Kuczma, Functional Equations in a Single Variable, Polish Scientific Publishers, Warszawa, 1968.  Google Scholar

[9]

E. Liz, Stability of non-autonomous difference equations: Simple ideas leading to useful results, J. Difference Eq. Appl., 17 (2011), 203-220.  doi: 10.1080/10236198.2010.549007.  Google Scholar

[10]

J. Matkowski and W. Zhang, Method of characteristic for functional equations in polynomial form, Acta Math. Sinica, New Series, 13 (1997), 421-432.  doi: 10.1007/BF02560023.  Google Scholar

[11]

A. Mukherjea and J. S. Ratti, On a functional equation involving iterates of a bijection on the unit interval, Nonlinear Anal., 7 (1983), 899-908.  doi: 10.1016/0362-546X(83)90065-2.  Google Scholar

[12]

Ch. G. Philos and I. K. Purnaras, On non-autonomous linear difference equations with continuous variable, J. Difference Eq. Appl., 12 (2006), 651-668.  doi: 10.1080/10236190600652360.  Google Scholar

[13]

H. Sedaghat, Global attractivity in a class of non-autonomous, nonlinear, higher order difference equations, J. Difference Eq. Appl., 19 (2013), 1049-1064.  doi: 10.1080/10236198.2012.707196.  Google Scholar

[14]

J. Tabor and M. Żołdak, Iterative equations in Banach spaces, J. Math. Anal. Appl., 299 (2004), 651-662.  doi: 10.1016/j.jmaa.2004.06.011.  Google Scholar

[15]

X. Tang and W. Zhang, Continuous solutions of a second iterative equation, Publ. Math. Debrecen, 93 (2018), 303-321.   Google Scholar

[16]

W. Zhang, Discussion on the iterated equation $\sum_{i=1}^{n}\lambda_if^i(x) =F(x)$, Chinese Sci. Bull., 32 (1987), 1444-1451.   Google Scholar

[17]

W. Zhang, Stability of the solution of the iterated equation $\sum_{i=1}^{n}\lambda_if^i(x)=F(x)$, Acta Math. Sci. English Series, 8 (1988), 421-424.  doi: 10.1016/S0252-9602(18)30318-7.  Google Scholar

[18]

W. Zhang, Discussion on the differentiable solutions of the iterated equation $\sum_{i=1}^n \lambda_i f^i(x)= F(x)$, Nonlinear Anal., 15 (1990), 387-398.  doi: 10.1016/0362-546X(90)90147-9.  Google Scholar

[19]

W. Zhang and J. A. Baker, Continuous solutions of a polynomial-like iterative equation with variable coefficients, Ann. Polon. Math., 73 (2000), 29-36.  doi: 10.4064/ap-73-1-29-36.  Google Scholar

[20]

W. ZhangK. Nikodem and B. Xu, Convex solutions of polynomial-like iterative equations, J. Math. Anal. Appl., 315 (2006), 29-40.  doi: 10.1016/j.jmaa.2005.10.006.  Google Scholar

Figure 1.  Graph of $ \alpha_{\sigma(1)} $ in (35)
Figure 2.  Graph of $ \varphi $ in Example 1
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