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Conditional equations for quadratic functions

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Abstract

We consider quadratic functions f that satisfy the additional equation y2 f(x) =  x2 f(y) for the pairs \({ (x,y) \in \mathbb{R}^2}\) that fulfill the condition P(x, y) =  0 for some fixed polynomial P of two variables. If P(x, y) =  ax +  by +  c with \({ a , b , c \in \mathbb{R}}\) and \({(a^2 + b^2)c \neq 0}\) or P(x,y) =  xn − y with a natural number \({n \geq 2}\), we prove that f(x) =  f(1) x2 for all \({x \in \mathbb{R}}\). Some related problems, admitting quadratic functions generated by derivations, are considered as well.

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References

  1. Aczél J.: Some unsolved problems in the theory of functional equations. Arch. Math., 15, 435–444 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  2. J. Aczél, The general solution of two functional equations by reduction to functions additive in two variables and with the aid of Hamel bases, Glasnik Mat.-Fiz. Astronom. Ser. II Društvo Mat. Fiz. Hrvatske, 20 (1965), 65–73.

  3. J. Aczél and J. Dhombres, Functional Equations in Several Variables, Encyclopaedia of Mathematics and its Applications, vol. 31, Cambridge Univ. Press (Cambridge–New York–New Rochelle–Melbourne–Sydney, 1989).

  4. Benz W.: 5 Problem (in Report of Meeting: The twenty-seventh international symposium on functional equations). Aequationes Math., 39, 302 (1990)

    Google Scholar 

  5. Boros Z., Erdei P.: A conditional equation for additive functions. Aequationes Math., 70, 309–313 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  6. Ebanks B.: Characterizing ring derivations of all orders via functional equations: results and open problems. Aequationes Math., 89, 685–718 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ebanks B.: Polynomially linked additive functions. Aequationes Math., 91, 317–330 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  8. A. Grząślewicz, Some remarks to additive functions, Math. Japon., 23 (1978/79), 573–578.

  9. Gselmann E.: Notes on the characterization of derivations. Acta Sci. Math. (Szeged), 78, 137–145 (2012)

    MathSciNet  MATH  Google Scholar 

  10. Gselmann E.: Derivations and linear functions along rational functions. Monatsh. Math., 169, 355–370 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  11. E. Gselmann, Cs. Vincze and G. Kiss, On functional equations characterizing derivations: methods and examples, arXiv:1709.03038 [math.CA].

  12. Halter-Koch F.: Characterization of field homomorphisms and derivations by functional equations. Aequationes Math., 59, 298–305 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  13. Halter-Koch F.: A characterization of derivations by functional equations. Math. Pannon., 11, 187–190 (2000)

    MathSciNet  MATH  Google Scholar 

  14. F. Halter-Koch und L. Reich, Charakterisierung von Derivationen höherer Ordnung mittels Funktionalgleichungen, österreich. Akad. Wiss. Math.-Natur. Kl. Sitzungsber II, 207 (1998), 123–131.

  15. Jurkat W. B.: On Cauchy’s functional equation. Proc. Amer. Math. Soc., 16, 683–686 (1965)

    MathSciNet  MATH  Google Scholar 

  16. Pl. Kannappan and S. Kurepa, Some relations between additive functions I, Aequationes Math., 4 (1970), 163–175.

  17. Kannappan Pl., Kurepa S.: Some relations between additive functions II. Aequationes Math., 6, 46–58 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  18. M. Kuczma, An Introduction to the Theory of Functional Equations and Inequalities (second edition), Birkhäuser (Basel–Boston–Berlin, 2009).

  19. S. Kurepa, The Cauchy functional equation and scalar product in vector spaces, Glasnik Mat.-Fiz. Astronom. Ser. II, Društvo Mat. Fiz. Hrvatske, 19 (1964), 23–36.

  20. S. Kurepa, Remarks on the Cauchy functional equation, Publ. Inst. Math. (Beograd) (N.S.), 5(19) (1965), 85–88.

  21. Gy. Maksa, On the trace of symmetric bi-derivations, C. R. Math. Rep. Acad. Sci. Canada, IX (1987), 303–307.

  22. Nishiyama A., Horinouchi S.: On a system of functional equations. Aequationes Math., 1, 1–5 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  23. Reich L.: Derivationen zweiter Ordnung als Lösungen von Funktionalgleichungen—ein überblick. Grazer Math. Ber., 337, 45–65 (1998)

    MATH  Google Scholar 

  24. J. Unger und L. Reich, Derivationen höherer Ordnung als Lösungen von Funktionalgleichungen, Grazer Math. Ber., 336 (1998), 1–83.

  25. O. Zariski and P. Samuel, Commutative Algebra. Vol. I, with the cooperation of I. S. Cohen, The University Series in Higher Mathematics, D. Van Nostrand Company, Inc. (Princeton, NJ, 1958).

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Authors and Affiliations

  1. Institute of Mathematics, University of Debrecen, P. O. Box: 400, 4002, Debrecen, Hungary

    Z. Boros

  2. Department of Economic Sciences, Sapientia University, Miercurea-Ciuc, Romania

    E. Garda-Mátyás

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  1. Z. Boros
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  2. E. Garda-Mátyás
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Correspondence to Z. Boros.

Additional information

Research of Z. Boros has been supported by the Hungarian Scientific Research Fund (OTKA) grant K-111651.

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Boros, Z., Garda-Mátyás, E. Conditional equations for quadratic functions. Acta Math. Hungar. 154, 389–401 (2018). https://doi.org/10.1007/s10474-018-0795-x

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  • DOI: https://doi.org/10.1007/s10474-018-0795-x

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