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Iterative Roots of Non-PM Functions and Denseness

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Abstract

The characteristic interval plays a vital role on the existence of iterative roots of PM functions with height less than or equal to one. In this paper, we define the characteristic interval for continuous functions and prove theorems on extension and nonexistence of iterative roots for a class of continuous non-PM functions on a closed and bounded interval I. Also, we prove that a class of continuous non-PM functions, which do not possess any iterative roots, is dense in C(I, I).

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References

  1. Babbage, C.: An essay towards the calculus of functions. Philos. Trans. R. Soc. Lond. 105, 389–423 (1815)

    Article  Google Scholar 

  2. Blokh, A.M.: The set of all iterates is nowhere dense in \(C([0,1],[0,1])\). Trans. Am. Math. Soc. 333(2), 787–798 (1992)

    MathSciNet  MATH  Google Scholar 

  3. Bödewadt, U.T.: Zur iteration reeller funktionen. Math. Z. 49, 497–516 (1944)

    Article  MathSciNet  MATH  Google Scholar 

  4. Castillo, E., Iglesias, A., Ruiz-Cobo, R.: Functional Equations in Applied Sciences, vol. 199. Elsevier, Amsterdam (2004)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  6. Haídukov, P.I.: On searching a function from a given iterate. Uč Zap. Buriatsk. Ped. Inst. 15, 3–28 (1958)

    Google Scholar 

  7. Kuczma, M.: On the functional equation \(\varphi ^{n}(x)=g(x)\). Ann. Pol. Math. 11, 161–175 (1961)

    Article  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  9. Li, L., Chen, J.: Iterative roots of piecewise monotonic functions with finite nonmonotonicity height. J. Math. Anal. Appl. 411, 395–404 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  10. Liu, L., Jarczyk, W., Li, L., Zhang, W.: Iterative roots of piecewise monotonic functions of nonmonotonicity height not less than 2. Nonlinear Anal. 75(1), 286–303 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  11. Liu, L., Zhang, W.: Non-monotonic iterative roots extended from characteristic intervals. J. Math. Anal. Appl. 378(1), 359–373 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  12. Yinhe, L.: Existence of iterative roots for the sickle-like functions. J. Inequal. Appl. 2014, 204 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  13. Yinhe, L., Yingying, Z., Zhang, W.: Iterative roots of clenched single-plateau functions. Results Math. 71, 15–43 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  14. Zhang, W.: PM functions, their characteristic intervals and iterative roots. Ann. Pol. Math. 65(2), 119–128 (1997)

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics Education and RINS, Gyeongsang National University, Jinju, 52828, Korea

    Yeol Je Cho

  2. School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, 611731, Sichuan, China

    Yeol Je Cho

  3. Department of Mathematical and Computational Sciences, National Institute of Technology Karnataka, Surathkal, Mangaluru, 575025, India

    Suresh Kumar Murugan & Murugan Veerapazham

  4. Department of Mathematics, The Gandhigram Rural Institute - Deemed University, Dindigul, 624302, India

    Suresh Kumar Murugan

Authors
  1. Yeol Je Cho
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  2. Suresh Kumar Murugan
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  3. Murugan Veerapazham
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Correspondence to Murugan Veerapazham.

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Cho, Y.J., Murugan, S.K. & Veerapazham, M. Iterative Roots of Non-PM Functions and Denseness. Results Math 73, 13 (2018). https://doi.org/10.1007/s00025-018-0792-y

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  • DOI: https://doi.org/10.1007/s00025-018-0792-y

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