Solutions with big graph of iterative functional equations of the first order

Lech Bartłomiejczyk

Colloquium Mathematicae (1999)

  • Volume: 82, Issue: 2, page 223-230
  • ISSN: 0010-1354

Abstract

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We obtain a result on the existence of a solution with big graph of functional equations of the form g(x,𝜑(x),𝜑(f(x)))=0 and we show that it is applicable to some important equations, both linear and nonlinear, including those of Abel, Böttcher and Schröder. The graph of such a solution 𝜑 has some strange properties: it is dense and connected, has full outer measure and is topologically big.

How to cite

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Bartłomiejczyk, Lech. "Solutions with big graph of iterative functional equations of the first order." Colloquium Mathematicae 82.2 (1999): 223-230. <http://eudml.org/doc/210759>.

@article{Bartłomiejczyk1999,
abstract = {We obtain a result on the existence of a solution with big graph of functional equations of the form g(x,𝜑(x),𝜑(f(x)))=0 and we show that it is applicable to some important equations, both linear and nonlinear, including those of Abel, Böttcher and Schröder. The graph of such a solution 𝜑 has some strange properties: it is dense and connected, has full outer measure and is topologically big.},
author = {Bartłomiejczyk, Lech},
journal = {Colloquium Mathematicae},
keywords = {iterative functional equation; big graph; second category; Haar zero set; periodic points; topological spaces; Cauchy functional equation},
language = {eng},
number = {2},
pages = {223-230},
title = {Solutions with big graph of iterative functional equations of the first order},
url = {http://eudml.org/doc/210759},
volume = {82},
year = {1999},
}

TY - JOUR
AU - Bartłomiejczyk, Lech
TI - Solutions with big graph of iterative functional equations of the first order
JO - Colloquium Mathematicae
PY - 1999
VL - 82
IS - 2
SP - 223
EP - 230
AB - We obtain a result on the existence of a solution with big graph of functional equations of the form g(x,𝜑(x),𝜑(f(x)))=0 and we show that it is applicable to some important equations, both linear and nonlinear, including those of Abel, Böttcher and Schröder. The graph of such a solution 𝜑 has some strange properties: it is dense and connected, has full outer measure and is topologically big.
LA - eng
KW - iterative functional equation; big graph; second category; Haar zero set; periodic points; topological spaces; Cauchy functional equation
UR - http://eudml.org/doc/210759
ER -

References

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  1. [1] L. Bartłomiejczyk, Solutions with big graph of homogeneous functional equations in a single variable, Aequationes Math. 56 (1998), 149-156. Zbl0913.39014
  2. [2] L. Bartłomiejczyk, Solutions with big graph of the equation of invariant curves, submitted. Zbl0993.39015
  3. [3] L. Bartłomiejczyk, Iterative roots with big graph, submitted. Zbl1014.39500
  4. [4] L. Bartłomiejczyk, Solutions with big graph of an equation of the second iteration, submitted. 
  5. [5] J. P. R. Christensen, On sets of Haar measure zero in abelian Polish groups, Israel J. Math. 13 (1972), 255-260. 
  6. [6] J. P. R. Christensen, Topology and Borel Structure, North-Holland Math. Stud. 10, North-Holland, Amsterdam, 1974. 
  7. [7] P. R. Halmos, Measure Theory, Grad. Texts in Math. 18, Springer, New York, 1974. 
  8. [8] F. B. Jones, Connected and disconnected plane sets and the functional equation f(x)+f(y)=f(x+y), Bull. Amer. Math. Soc. 48 (1942), 115-120. Zbl0063.03063
  9. [9] P. Kahlig and J. Smítal, On the solutions of a functional equation of Dhombres, Results Math. 27 (1995), 362-367. Zbl0860.39030
  10. [10] M. Kuczma, Functional Equations in a Single Variable, Monografie Mat. 46, PWN-Polish Sci. Publ., Warszawa, 1968. Zbl0196.16403
  11. [11] M. Kuczma, An Introduction to the Theory of Functional Equations and Inequalities. Cauchy's Equation and Jensen's Inequality, Prace Nauk. Uniw. Śląskiego 489, PWN & Uniw. Śląski, Warszawa-Kraków-Katowice, 1985. 
  12. [12] M. Kuczma, B. Choczewski and R. Ger, Iterative Functional Equations, Encyclopedia Math. Appl. 32, Cambridge Univ. Press, Cambridge, 1990. Zbl0703.39005
  13. [13] W. Kulpa, On the existence of maps having graphs connected and dense, Fund. Math. 76 (1972), 207-211. Zbl0245.54012
  14. [14] K. Kuratowski and A. Mostowski, Set Theory, Stud. Logic Found. Math. 86, PWN and North-Holland, Warszawa-Amsterdam, 1976. 
  15. [15] K. R. Parthasarathy, Probability Measures on Metric Spaces, Academic Press, New York, 1967. Zbl0153.19101
  16. [16] Gy. Targonski, Topics in Iteration Theory, Vandenhoeck & Ruprecht, Göttingen, 1981. Zbl0454.39003

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