If a Set is Unbounded It Has an Unbounded Continuous Function

Abstract

Motivated by nonautonomous difference equations, we study a functional equation with nonautonomous iteration of order n for bounded solutions and unbounded solutions. We give a condition for the existence of bounded solutions in the case that a given function is bounded and a condition for the existence of unbounded solutions in the case that a given function is unbounded. We also discuss the existence of unbounded solutions in the case that a given function is bounded.

References

1. Blokh, A.M.: The set of all iterates is nowhere dense in \(C([0,1],[0,1])\). Trans. Am. Math. Soc. 333, 787–798 (1992)

   MathSciNet  MATH  Google Scholar

2. Brillouët-Belluot, N.: Problem 15, Proceedings of 38th ISFE (2000 Hungary). Aequ. Math. 61, 304 (2001)

   Google Scholar

3. Brillouët-Belluot, N., Zhang, W.: On a class of iterative-difference equations. J. Differ. Equ. Appl. 16(11), 1237–1255 (2010)

   Article  MathSciNet  Google Scholar

4. Fort Jr., M.K.: The embedding of homeomorphisms in flows. Proc. Am. Math. Soc. 6, 960–967 (1955)

   Article  MathSciNet  Google Scholar

5. Geiselhart, R., Wirth, F.: Solving iterative functional equations for a class of piecewise linear \({\cal{K}}_{\infty }\)-functions. J. Math. Anal. Appl. 411, 652–664 (2014)

   Article  MathSciNet  Google Scholar

6. Humke, P.D., Laczkovich, M.: The Borel structure of iterates of continuous functions. Proc. Edinb. Math. Soc. 32, 483–494 (1989)

   Article  MathSciNet  Google Scholar

7. Isaacs, R.: Iterates of fractional order. Can. J. Math. 2, 409–416 (1950)

   Article  MathSciNet  Google Scholar

8. Jarczyk, W.: On an equation of linear iteration. Aequ. Math. 51, 303–310 (1996)

   Article  MathSciNet  Google Scholar

9. Karlin, S., McGregor, J.: Embedding iterates of analytic functions with two fixed points into continuous groups. Trans. Am. Math. Soc. 132, 137–145 (1968)

   Article  MathSciNet  Google Scholar

10. Krassowska, D., Zdun, M.C.: On iteration groups of \(C^1\)-diffeomorphisms with two fixed points. J. Math. Anal. Appl. 382, 204–213 (2011)

    Article  MathSciNet  Google Scholar

11. Kuczma, M., Choczewski, B., Ger, R.: Iterative Functional Equations. Cambridge University Press, Cambridge (1990)

    Book  Google Scholar

12. Liu, L., Jarczyk, W., Li, L., Zhang, W.: Iterative roots of piecewise monotonic functions of nonmonotonicity height not less than 2. Nonlinear Anal. 75, 286–303 (2012)

    Article  MathSciNet  Google Scholar

13. Liu, J., Yu, Z., Zhang, W.: Invariant curves for a second-order difference equation modelled from macroeconomics. J. Differ. Equ. Appl. 21, 757–773 (2015)

    Article  MathSciNet  Google Scholar

14. Matkowski, J., Zhang, W.: Method of characteristic for functional equations in polynomial form. Acta Math. Sin. New Ser. 13(3), 421–432 (1997)

    Article  MathSciNet  Google Scholar

15. Mukherjea, A., Ratti, J.S.: On a functional equation involving iterates of a bijection on the unit interval. Nonlinear Anal. 7, 899–908 (1983)

    Article  MathSciNet  Google Scholar

16. Ng, C.T., Zhang, W.: Invariant curves for planar mappings. J. Differ. Equ. Appl. 3, 147–168 (1997)

    Article  MathSciNet  Google Scholar

17. Rice, R.E., Schweizer, B., Sklar, A.: When is \(f (f (z)) = az^2 + bz + c\)? Am. Math. Mon. 87, 252–263 (1980)

    MATH  Google Scholar

18. Tabor, J., Tabor, J.: On a linear iterative equation. Results Math. 27, 412–421 (1995)

    Article  MathSciNet  Google Scholar

19. Tabor, J., Żołdak, M.: Iterative equations in Banach spaces. J. Math. Anal. Appl. 299, 651–662 (2004)

    Article  MathSciNet  Google Scholar

20. Tang, X., Zhang, W.: Continuous solutions of a second order iterative equation. Publ. Math. Debr. 93(3–4), 303–321 (2018)

    Article  MathSciNet  Google Scholar

21. Targonski, G.: Topics in Iteration Theory. Vandenhoeck and Ruprecht, Göttingen (1981)

    MATH  Google Scholar

22. Zdun, M.C.: Continuous and Differentiable Iteration Semigroups. Silesian Univ. Press, Katowice (1979)

    MATH  Google Scholar

23. Zdun, M.C., Zhang, W.: Koenigs embedding flow problem with global \(C^1\) smoothness. J. Math. Anal. Appl. 374, 633–643 (2011)

    Article  MathSciNet  Google Scholar

24. Zeng, Y., Zhang, W.: Continuous solutions of an iterative-difference equation and Brillouët-Belluot's problem. Publ. Math. Debr. 78(3–4), 613–624 (2011)

    Article  Google Scholar

25. Zhang, J., Yang, L.: Discussion on iterative roots of piecewise monotone functions. Acta Math. Sin. 26, 398–412 (1983). (in Chinese)

    MATH  Google Scholar

26. Zhang, W.: Discussion on the differentiable solutions of the iterated equation \( \sum _{i=1}^n \lambda _i f^i(x)= F(x) \). Nonlinear Anal. 15, 387–398 (1990)

    Article  MathSciNet  Google Scholar

27. Zhang, W., Nikodem, K., Xu, B.: Convex solutions of polynomial-like iterative equations. J. Math. Anal. Appl. 315, 29–40 (2006)

    Article  MathSciNet  Google Scholar

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Authors and Affiliations

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

   Xiao Tang

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

   Weinian Zhang

Corresponding author

Correspondence to Weinian Zhang.

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Supported by NSFC # 11771307, NSFC # 11726623, and NSFC # 11831012.

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Tang, X., Zhang, W. Bounded or unbounded solutions of a functional equation with nonautonomous iteration. Aequat. Math. 94, 259–275 (2020). https://doi.org/10.1007/s00010-019-00697-y

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• Received: 29 March 2019

• Revised: 18 December 2019

• Published: 03 January 2020

• Issue Date: April 2020

• DOI : https://doi.org/10.1007/s00010-019-00697-y

Keywords

• Nonautonomous iteration
• Bounded solution
• Unbounded solution
• Lipschitz condition
• Contraction

Mathematics Subject Classification

• 39B12
• 37E05

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