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Analytic Number Theory, Approximation Theory, and Special Functions pp 759–772Cite as

Orthogonally Additive: Additive Functional Equation

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Abstract

Using fixed point method, we prove the Hyers–Ulam stability of the orthogonally additive–additive functional equation

$$\displaystyle{f\left (\frac{x} {2} + y\right ) + f\left (\frac{x} {2} + z\right ) = f(x) + f(y) + f(z)}$$

for all x, y, z with x ⊥ y, in orthogonality Banach spaces and in non-Archimedean orthogonality Banach spaces.

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References

  1. Alonso, J., Benítez, C.: Orthogonality in normed linear spaces: a survey I. Main properties. Extracta Math. 3, 1–15 (1988)

    Google Scholar 

  2. Alonso, J., Benítez, C.: Orthogonality in normed linear spaces: a survey II. Relations between main orthogonalities. Extracta Math. 4, 121–131 (1989)

    Google Scholar 

  3. Birkhoff, G.: Orthogonality in linear metric spaces. Duke Math. J. 1, 169–172 (1935)

    Article  MathSciNet  Google Scholar 

  4. Cădariu, L., Radu, V.: Fixed points and the stability of Jensen’s functional equation. J. Inequal. Pure Appl. Math. 4(1), Art. ID 4 (2003)

    Google Scholar 

  5. Cădariu, L., Radu, V.: On the stability of the Cauchy functional equation: a fixed point approach. Grazer Math. Ber. 346, 43–52 (2004)

    MATH  Google Scholar 

  6. Cădariu, L., Radu, V.: Fixed point methods for the generalized stability of functional equations in a single variable. Fixed Point Theory Appl. 2008, Art. ID 749392 (2008).

    Google Scholar 

  7. Carlsson, S.O.: Orthogonality in normed linear spaces. Ark. Mat. 4, 297–318 (1962)

    Article  MATH  MathSciNet  Google Scholar 

  8. Cholewa, P.W.: Remarks on the stability of functional equations. Aequationes Math. 27, 76–86 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  9. Czerwik, S.: On the stability of the quadratic mapping in normed spaces. Abh. Math. Sem. Univ. Hamburg 62, 59–64 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  10. Czerwik, S.: Functional Equations and Inequalities in Several Variables. World Scientific Publishing Company, New Jersey (2002)

    Book  MATH  Google Scholar 

  11. Czerwik, S.: Stability of Functional Equations of Ulam–Hyers–Rassias Type. Hadronic Press, Palm Harbor (2003)

    Google Scholar 

  12. Deses, D.: On the representation of non-Archimedean objects. Topology Appl. 153, 774–785 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  13. Diaz, J., Margolis, B.: A fixed point theorem of the alternative for contractions on a generalized complete metric space. Bull. Amer. Math. Soc. 74, 305–309 (1968)

    Article  MATH  MathSciNet  Google Scholar 

  14. Diminnie, C.R.: A new orthogonality relation for normed linear spaces. Math. Nachr. 114, 197–203 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  15. Drljević, F.: On a functional which is quadratic on A-orthogonal vectors. Publ. Inst. Math. (Beograd) 54, 63–71 (1986)

    Google Scholar 

  16. Eshaghi Gordji, M.: A characterization of (σ, τ)-derivations on von Neumann algebras. Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 73(1), 111–116 (2011)

    MATH  MathSciNet  Google Scholar 

  17. Eshaghi Gordji, M., Bodaghi, A., Park, C.: A fixed point approach to the stability of double Jordan centralizers and Jordan multipliers on Banach algebras. Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 73(2), 65–74 (2011)

    MATH  MathSciNet  Google Scholar 

  18. Eshaghi Gordji, M., Khodaei, H., Khodabakhsh, R.: General quartic-cubic-quadratic functional equation in non-Archimedean normed spaces. Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 72(3), 69–84 (2010)

    MATH  MathSciNet  Google Scholar 

  19. Eshaghi Gordji, M., Savadkouhi, M.B.: Approximation of generalized homomorphisms in quasi-Banach algebras. An. Stiint. Univ. Ovidius Constanta Ser. Mat. 17(2), 203–213 (2009)

    MATH  MathSciNet  Google Scholar 

  20. Fochi, M.: Functional equations in A-orthogonal vectors. Aequationes Math. 38, 28–40 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  21. Ger, R., Sikorska, J.: Stability of the orthogonal additivity. Bull. Polish Acad. Sci. Math. 43, 143–151 (1995)

    MATH  MathSciNet  Google Scholar 

  22. Gudder, S., Strawther, D.: Orthogonally additive and orthogonally increasing functions on vector spaces. Pacific J. Math. 58, 427–436 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  23. Hensel, K.: Ubereine news Begrundung der Theorie der algebraischen Zahlen. Jahresber. Deutsch. Math. Verein 6, 83–88 (1897)

    Google Scholar 

  24. Hyers, D.H.: On the stability of the linear functional equation. Proc. Nat. Acad. Sci. U.S.A. 27, 222–224 (1941)

    Article  MathSciNet  Google Scholar 

  25. Hyers, D.H., Isac, G., Rassias, Th.M.: Stability of Functional Equations in Several Variables. Birkh\(\ddot{a}\) user, Basel (1998)

    Google Scholar 

  26. Isac, G., Rassias, Th.M.: Stability of ψ-additive mappings: Applications to nonlinear analysis. Internat. J. Math. Math. Sci. 19, 219–228 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  27. James, R.C.: Orthogonality in normed linear spaces. Duke Math. J. 12, 291–302 (1945)

    Article  MATH  MathSciNet  Google Scholar 

  28. James, R.C.: Orthogonality and linear functionals in normed linear spaces. Trans. Amer. Math. Soc. 61, 265–292 (1947)

    Article  MathSciNet  Google Scholar 

  29. Jung, S.: Hyers–Ulam–Rassias Stability of Functional Equations in Mathematical Analysis. Hadronic Press, Palm Harbor (2001)

    MATH  Google Scholar 

  30. Jung, Y., Chang, I.: The stability of a cubic type functional equation with the fixed point alternative. J. Math. Anal. Appl. 306, 752–760 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  31. Katsaras, A.K., Beoyiannis, A.: Tensor products of non-Archimedean weighted spaces of continuous functions. Georgian Math. J. 6, 33–44 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  32. Khrennikov, A.: Non-Archimedean analysis: quantum paradoxes, dynamical systems and biological models. In: Mathematics and its Applications, vol. 427. Kluwer Academic Publishers, Dordrecht (1997)

    Google Scholar 

  33. Miheţ, D., Radu, V.: On the stability of the additive Cauchy functional equation in random normed spaces. J. Math. Anal. Appl. 343, 567–572 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  34. Mirzavaziri, M., Moslehian, M.S.: A fixed point approach to stability of a quadratic equation. Bull. Braz. Math. Soc. 37, 361–376 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  35. Moslehian, M.S.: On the orthogonal stability of the Pexiderized quadratic equation. J. Difference Equat. Appl. 11, 999–1004 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  36. Moslehian, M.S.: On the stability of the orthogonal Pexiderized Cauchy equation. J. Math. Anal. Appl. 318, 211–223 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  37. Moslehian, M.S., Rassias, Th.M.: Orthogonal stability of additive type equations. Aequationes Math. 73, 249–259 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  38. Moslehian, M.S., Sadeghi, Gh.: A Mazur-Ulam theorem in non-Archimedean normed spaces. Nonlinear Anal. TMA 69, 3405–3408 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  39. Nyikos, P.J.: On some non-Archimedean spaces of Alexandrof and Urysohn. Topology Appl. 91, 1–23 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  40. Paganoni, L., Rätz, J.: Conditional function equations and orthogonal additivity. Aequationes Math. 50, 135–142 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  41. Park, C.: Fixed points and Hyers–Ulam–Rassias stability of Cauchy–Jensen functional equations in Banach algebras. Fixed Point Theory Appl. 2007, Art. ID 50175 (2007)

    Google Scholar 

  42. Park, C.: Generalized Hyers–Ulam–Rassias stability of quadratic functional equations: a fixed point approach. Fixed Point Theory Appl. 2008, Art. ID 493751 (2008)

    Google Scholar 

  43. Park, C., Park, J.: Generalized Hyers–Ulam stability of an Euler–Lagrange type additive mapping. J. Difference Equat. Appl. 12, 1277–1288 (2006)

    Article  MATH  Google Scholar 

  44. Pinsker, A.G.: Sur une fonctionnelle dans l’espace de Hilbert. C. R. (Dokl.) Acad. Sci. URSS, n. Ser. 20, 411–414 (1938)

    Google Scholar 

  45. Radu, V.: The fixed point alternative and the stability of functional equations. Fixed Point Theory 4, 91–96 (2003)

    MATH  MathSciNet  Google Scholar 

  46. Rassias, Th.M.: On the stability of the linear mapping in Banach spaces. Proc. Amer. Math. Soc. 72, 297–300 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  47. Rassias, Th.M.: On the stability of the quadratic functional equation and its applications. Studia Univ. Babeş-Bolyai Math. 43, 89–124 (1998)

    MATH  Google Scholar 

  48. Rassias, Th.M.: The problem of S.M. Ulam for approximately multiplicative mappings. J. Math. Anal. Appl. 246, 352–378 (2000)

    Google Scholar 

  49. Rassias, Th.M.: On the stability of functional equations in Banach spaces. J. Math. Anal. Appl. 251, 264–284 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  50. Rassias, Th.M. (ed.): Functional Equations, Inequalities and Applications. Kluwer Academic Publishers, Dordrecht (2003)

    MATH  Google Scholar 

  51. Rätz, J.: On orthogonally additive mappings. Aequationes Math. 28, 35–49 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  52. Rätz, J., Szabó, Gy.: On orthogonally additive mappings IV. Aequationes Math. 38, 73–85 (1989)

    Google Scholar 

  53. Skof, F.: Proprietà locali e approssimazione di operatori. Rend. Sem. Mat. Fis. Milano 53, 113–129 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  54. Sundaresan, K.: Orthogonality and nonlinear functionals on Banach spaces. Proc. Amer. Math. Soc. 34, 187–190 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  55. Szabó, Gy.: Sesquilinear-orthogonally quadratic mappings. Aequationes Math. 40, 190–200 (1990)

    Google Scholar 

  56. Ulam, S.M.: Problems in Modern Mathematics. Wiley, New York (1960)

    Google Scholar 

  57. Vajzović, F.: Über das Funktional H mit der Eigenschaft: \((x,y) = 0 \Rightarrow H(x + y) + H(x - y) = 2H(x) + 2H(y)\). Glasnik Mat. Ser. III 2(22), 73–81 (1967)

    Google Scholar 

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Acknowledgements

This work was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2009-0070788).

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  1. Department of Mathematics, Hanyang University, Seoul, 133-791, Republic of Korea

    Choonkil Park

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  1. Choonkil Park
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Correspondence to Choonkil Park .

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Editors and Affiliations

  1. Mathematical Institute, Serbian Academy of Sciences and Arts, Belgrade, Serbia

    Gradimir V. Milovanović

  2. Department of Mathematics, ETH Zürich, Zürich, Switzerland

    Michael Th. Rassias

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Dedicated to Professor Hari M. Srivastava

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Park, C. (2014). Orthogonally Additive: Additive Functional Equation. In: Milovanović, G., Rassias, M. (eds) Analytic Number Theory, Approximation Theory, and Special Functions. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-0258-3_29

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