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  December 2020 Early access  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 and Continuous Dynamical Systems, 2020, 40 (12) : 6967-6984. doi: 10.3934/dcds.2020214
References:
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R. Brück and M. Büger, Generalized iteration, Comput. Methods Funct. Theory, 3 (2003), 201-252.  doi: 10.1007/BF03321035.

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

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

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

[5]

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

[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.

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

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

[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.

[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.

[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.

[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.

[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.

[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.

[15]

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

[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. 

[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.

[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.

[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.

[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.

show all references

References:
[1]

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

[2]

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

[3]

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

[4]

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

[5]

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

[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.

[7]

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

[8]

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

[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.

[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.

[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.

[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.

[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.

[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.

[15]

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

[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. 

[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.

[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.

[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.

[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.

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