The general linear equation on open connected sets

Abstract

Fix non-zero reals \(\alpha _1\), \(\ldots , \)\(\alpha _n\) with \(n\ge 2\) and let \(K\) be a non-empty open connected set in a topological vector space such that \(\sum _{i\le n}\alpha _iK\subseteq K\) (which holds, in particular, if \(K\) is an open convex cone and \(\alpha _1,\ldots ,\alpha _n>0\)). Let also \(Y\) be a vector space over \(\mathbb{F}\it :=\mathbb{Q} \it (\alpha _1,\ldots ,\alpha _n)\). We show, among others, that a function \(f : K\rightarrow Y\) satisfies the general linear equation

$$\forall x_1,\ldots, x_n \in K, \quad f\big (\sum _{i\le n}\alpha _i x_i\big )=\sum _{i\le n}\alpha _i f(x_i)$$

if and only if there exist a unique \(\mathbb{F}\it \)-linear \(A X \rightarrow Y\) and unique \(b\in Y\) such that \(f(x)=A(x)+b\) for all \(x \in K\), with \(b=0\) if \(\sum _{i\le n}\alpha _i\ne 1\). The main tool of the proof is a general version of a result Radó and Baker on the existence and uniqueness of extension of the solution on the classical Pexider equation.

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References

  1. 1.

    J. Aczél, A Short Course on Functional Equations, Theory and Decision Library. Series B: Mathematical and Statistical Methods, D. Reidel Publishing Co. (Dordrecht, 1987)

  2. 2.

    Aczél, J.: Extension of a generalized Pexider equation. Proc. Amer. Math. Soc. 133, 3227–3233 (2005)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Chudziak, J., Tabor, J.: Generalized Pexider equation on a restricted domain. J. Math. Psych. 52, 389–392 (2008)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Chudziak, M., Sobek, B.: Generalized Pexider equation on an open domain. Results Math. 71, 1359–1372 (2017)

    MathSciNet  Article  Google Scholar 

  5. 5.

    G. L. Forti and L. Paganoni, \(\Omega \)-additive functions on topological groups, in: Constantin Carathéodory: an International Tribute, Vols. I,II, World Sci. Publ. (Teaneck, NJ, 1991), pp. 312–330

  6. 6.

    Głazowska, D., Leonetti, P., Matkowski, J., Tringali, S.: Commutativity of integral quasi-arithmetic means on measure spaces. Acta Math. Hungar. 153, 350–355 (2017)

    MathSciNet  Article  Google Scholar 

  7. 7.

    E. Gselmann, G. Kiss, and C. Vincze, On a class of linear functional equations without range condition, arXiv:1903.07974

  8. 8.

    Kuczma, M.: Functional equations on restricted domains. Aequationes Math. 18, 1–34 (1978)

    MathSciNet  Article  Google Scholar 

  9. 9.

    M. Kuczma, An Introduction to the Theory of Functional Equations and Inequalities. Cauchy's Equation and Jensen's Inequality, 2nd ed., Edited by Attila Gilányi, Birkhäuser Verlag (Basel, 2009).

  10. 10.

    P. Leonetti, J. Matkowski, and S. Tringali, On the commutation of generalized means on probability spaces, Indag. Math. (N.S.), 27 (2016), 945–953

  11. 11.

    Páles, Z.: Extension theorems for functional equations with bisymmetric operations. Aequationes Math. 63, 266–291 (2002)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Prager, W., Schwaiger, J.: The inhomogeneous general linear functional equation. Aequationes Math. 89, 1167–1187 (2015)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Radó, F., Baker, J.A.: Pexider's equation and aggregation of allocations. Aequationes Math. 32, 227–239 (1987)

    MathSciNet  Article  Google Scholar 

  14. 14.

    W. Rudin, Functional Analysis, 2nd ed., International Series in Pure and Applied Mathematics, McGraw-Hill, Inc. (New York, 1991)

  15. 15.

    Székelyhidi, L.: The general representation of an additive function on an open point set, Magyar Tud. Akad. Mat. Fiz. Oszt. Közl. 21, 503–509 (1973)

    MathSciNet  Google Scholar 

  16. 16.

    Székelyhidi, L.: On a class of linear functional equations, Publ. Math. Debrecen 29, 19–28 (1982)

    MathSciNet  MATH  Google Scholar 

  17. 17.

    Székelyhidi, L.: On a linear functional equation. Aequationes Math. 38, 113–122 (1989)

    MathSciNet  Article  Google Scholar 

  18. 18.

    L. Székelyhidi, Convolution Type Functional Equations on Topological Abelian Groups, World Scientific Publishing Co., Inc. (Teaneck, NJ, 1991).

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Acknowledgement

The authors are grateful to the anonymous referee for suggestions that helped improving the overall presentation of the article.

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Correspondence to P. Leonetti.

Additional information

P.L. was supported by the Austrian Science Fund (FWF), project F5512-N26.

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Leonetti, P., Schwaiger, J. The general linear equation on open connected sets. Acta Math. Hungar. 161, 201–211 (2020). https://doi.org/10.1007/s10474-019-00987-6

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Key words and phrases

  • Pexider equation
  • general linear equation
  • existence and uniqueness of extension
  • open connected set

Mathematics Subject Classification

  • primary 39B52
  • 15A06
  • secondary 39B22
  • 39B32