Journal of the Chungcheong Mathematical Society (충청수학회지)

The Chungcheong Mathematical Society (충청수학회)
권호 표지
DOI QR코드

DOI QR Code

REMARKS ON THE PAPER: ORTHOGONALLY ADDITIVE AND ORTHOGONALLY QUADRATIC FUNCTIONAL EQUATION

  • Received : 2013.02.09
  • Accepted : 2013.04.04
  • Published : 2013.05.15
Download PDF
⟨ Previous Next ⟩

Abstract

The main goal of this paper is to present the additional stability results of the following orthogonally additive and orthogonally quadratic functional equation $$f(\frac{x}{2}+y)+f(\frac{x}{2}-y)+f(\frac{x}{2}+z)+f(\frac{x}{2}-z)=\frac{3}{2}f(x)-\frac{1}{2}f(-x)+f(y)+f(-y)+f(z)+f(-z)$$ for all $x,y,z$ with $x{\bot}y$, which has been introduced in the paper [11], in orthogonality Banach spaces and in non-Archimedean orthogonality Banach spaces.

Keywords

References

  1. G. Birkhoff, Orthogonality in linear metric spaces, Duke. Math. J. 1 (1935), 169-172. https://doi.org/10.1215/S0012-7094-35-00115-6
  2. D. Deses, On the representation of non-Archimedean objects, Topology Appl. 153 (2005), 774-785. https://doi.org/10.1016/j.topol.2005.01.010
  3. F. Drljevic, On a functional which is quadratic on A-orthogonal vectors, Publ. Inst. Math. (Beograd) 54 (1986), 63-71.
  4. M. Fochi, Functional equations in A-orthogonal vectors, Aequations Math. 38 (1989), 28-40 https://doi.org/10.1007/BF01839491
  5. R. Ger and J. Sikorska, Stability of the orthogonal additivity, Bull. Plish Acad. Sci. Math. 43 (1995), 143-151.
  6. S. Gudder and D. Strawther, Orthogonally additive and orthogonally increasing functions on vector spaces, Pacific J. Math. 58 (1975), 427-436. https://doi.org/10.2140/pjm.1975.58.427
  7. K. Hensel, Ubereine news Begrundung der Theorie der algebraischen Zahlen, Jahresber. Deutsch. Math. Verein. 6 (1897), 83-88.
  8. R. C. James, Orthogonality in normed linear spaces, Trans. Amer. Math. Soc. 61 (1947), 265-292. https://doi.org/10.1090/S0002-9947-1947-0021241-4
  9. A. K. Katsaras and A. Beoyiannis, Tensor products of non-Archimedean weighted spaces of continuous functions, Georgian Math. J. 6 (1999), 33-44. https://doi.org/10.1023/A:1022926309318
  10. A. Khrennikov, Non-Archimedean analysis: quantum paradoxes, dynamical sys- tems and biological models, Mathematics and its Applications, Kluwer Academic Publishers Dordrecht 427 (1997).
  11. J. Lee, S. Lee, and C. Park, Orthogonally additive and orthogonally quadratic functional equation, Korean J. Math. to be appeared.
  12. M. S. Moslehian, On the orthogonal stability of the Pexiderized quadratic equation, J. Difference Equat. Appl. 11 (2005), 999-1004. https://doi.org/10.1080/10236190500273226
  13. M. S. Moslehian, On the orthogonal stability of the Pexiderized Cauchy equation, J. Math. Anal. Appl. 318 (2006), 211-223. https://doi.org/10.1016/j.jmaa.2005.05.052
  14. M. S. Moslehian and Th. M. Rassias, Orthogonal stability of additive type equations, Aequations Math. 73 (2007), 249-259. https://doi.org/10.1007/s00010-006-2868-0
  15. P. J. Nyikos, On some non-Archimedean spaces of Alexandrof and Urysohn, Topology Appl. 91 (1999), 1-23. https://doi.org/10.1016/S0166-8641(97)00239-3
  16. A. G. Pinsker, Sur une fonctionnelle dans l'espace de Hilbert, C. R. (Dokl.) Acad. Sci. URSS, n. Ser. 20 (1938), 411-414.
  17. L. Paganoni and J. Ratz, Conditional function equations and orthogonal additivity, Aequations Math. 50 (1995), 135-142. https://doi.org/10.1007/BF01831116
  18. J. Ratz, On orthogonally additive mappings, Aequations Math. 28 (1985), 35-49. https://doi.org/10.1007/BF02189390
  19. J. Ratz and Gy. Szabo, On orthogonally additive mappings IV, Aequations Math. 38 (1989), 73-85. https://doi.org/10.1007/BF01839496
  20. K. Sundaresan, Orthogonality and nonlinear functionals on Banach spaces, Proc. Amer. Math. Soc. 34 (1972), 187-190. https://doi.org/10.1090/S0002-9939-1972-0291835-X
  21. Gy. Szabo, Sesquilinear-orthogonally quadratic mappings, Aequations Math. 40 (1990), 190-200. https://doi.org/10.1007/BF02112295
  22. F. Vajzovic, Uber das Funcktional H mit der Eigenschaft: (x; y) = 0 ${\Rightarrow}$ H(x + y) + H(x - y) = 2H(x) + 2H(y), Glasnik Mat. Ser. III 2 22 (1967), 73-81.