Banach Journal of Mathematical Analysis

Conjugacy of P-configurations and nonlinear solutions to a certain conditional Cauchy equation

Orr Moshe Shalit

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Abstract

We study the connection between conjugations of a special kind of dynamical systems, called P-configurations, and solutions to homogeneous Cauchy type functional equations. We find that any two regular P-configurations are conjugate by a homeomorphism, but cannot be conjugate by a diffeomorphism. This leads us to the following conclusion (answering an open question posed by Paneah): there exist continuous nonlinear solutions to the functional equation: $$ f(t) = f\left(\frac{t+1}{2}\right) + f\left(\frac{t-1}{2}\right) \,\, , \,\, t \in [-1,1] . $$

Article information

Source
Banach J. Math. Anal. Volume 3, Number 1 (2009), 28-35.

Dates
First available: 21 April 2009

Permanent link to this document
http://projecteuclid.org/euclid.bjma/1240336420

Mathematical Reviews number (MathSciNet)
MR2461743

Zentralblatt MATH identifier
1157.39013

Subjects
Primary: 39B22: Equations for real functions [See also 26A51, 26B25]
Secondary: 37B99: None of the above, but in this section

Keywords
conditional functional equation Cauchy type functional equation P-configuration guided dynamical system

Citation

Shalit, Orr Moshe. Conjugacy of P-configurations and nonlinear solutions to a certain conditional Cauchy equation. Banach Journal of Mathematical Analysis 3 (2009), no. 1, 28--35. http://projecteuclid.org/euclid.bjma/1240336420.


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References

  • J. Aczél and J. Dhombres, Functional Equations in Several Variables, Cambridge University Press, 1989.
  • J. Dhombres and R. Ger, Conditional Cauchy equations, Glanik Mat. Ser. III, 13(33), (1978), no. 1, 39–62.
  • G.L. Forti, On some conditional Cauchy equations on thin sets, Boll. Un. Mat. Ital. B (6), 2 (1983), no. 1, 391–402.
  • W. Jarczyk, On continuous functions which are additive on their graphs, Selected topics in functional equations (Graz, 1986), Ber. No. 292, 66 pp., Ber. Math.-Statist. Sekt. Forschungsgesellsch. Joanneum, 285–-296, Forschungszentrum Graz, Graz, 1988.
  • M. Kuczma, Functional equations on restricted domains, Aequationes Math., 18 (1978), no. 1-2, 1–34.
  • J. Matkowski, Functions which are additive on their graphs and some generalizations, Rocznik Nauk.-Dydakt. Prace Mat. No. 13 (1993), 233–240.
  • B. Paneah, On the solvability of functional equations associated with dynamical systems with two generators, (Russian) Funktsional. Anal. i Prilozhen. 37 (2003), no. 1, 55–72, 96; translation in Funct. Anal. Appl. 37 (2003), no. 1, 46–60.
  • B. Paneah, Dynamic methods in the general theory of Cauchy type functional equations, Complex analysis and dynamical systems, 205–223, Contemp. Math., 364, Amer. Math. Soc., Providence, RI, 2004.
  • B. Paneah, On the over determinedness of some functional equations, Partial differential equations and applications. Discrete Contin. Dyn. Syst. 10 (2004), no. 1-2, 497–505.
  • M. Sablik, Some remarks on Cauchy equation on a curve, Demonstratio Math., 23 (1990), no. 2, 477–-490.
  • O.M. Shalit, Guided Dynamical Systems and Applications to Functional and Partial Differential Equations M.Sc. thesis, available at arXiv:math /0511638v2.
  • O.M. Shalit, On the overdeterminedness of a class of functional equations, Aequationes Math., 74 (2007), no. 3, 242–248.
  • M. Zdun, On the uniqueness of solutions of the functional equation $\varphi((x+f(x)) =\varphi(x)+\varphi(f(x))$, Aequationes Math., 8 (1972), 229–-232.