Korean Journal of Mathematics

The Kangwon-Kyungki Mathematical Society (강원경기수학회)
권호 표지
DOI QR코드

DOI QR Code

ORTHOGONALLY ADDITIVE AND ORTHOGONALLY QUADRATIC FUNCTIONAL EQUATION

  • Lee, Jung Rye (Department of Mathematics Daejin University) ;
  • Lee, Sung Jin (Department of Mathematics Daejin University) ;
  • Park, Choonkil (Department of Mathematics Research Institute for Natural Sciences Hanyang University)
  • Received : 2012.09.23
  • Accepted : 2013.01.10
  • Published : 2013.03.30
Download PDF
⟨ Previous Next ⟩

Abstract

Using the fixed point method, we prove the Ulam-Hyers stability of the orthogonally additive and orthogonally quadratic functional equation $$f(\frac{x}{2}+y)+f(\frac{x}{2}-y)+f(\frac{x}{2}+z)+f(\frac{x}{2}-z)=\frac{3}{2}f(x)-\frac{1}{2}f(-x)+f(y)+f(-y)+f(z)+f(-z)$$ (0.1) for all $x$, $y$, $z$ with $x{\bot}y$, in orthogonality Banach spaces and in non-Archimedean orthogonality Banach spaces.

Keywords

References

  1. J. Alonso and C. Benitez, Orthogonality in normed linear spaces: a survey I. Main properties, Extracta Math. 3 (1988), 1-15.
  2. J. Alonso and C. Benitez, Orthogonality in normed linear spaces: a survey II. Relations between main orthogonalities, Extracta Math. 4 (1989), 121-131.
  3. G. Birkhoff, Orthogonality in linear metric spaces, Duke Math. J. 1 (1935), 169-172. https://doi.org/10.1215/S0012-7094-35-00115-6
  4. L. Cadariu and V. Radu, Fixed points and the stability of Jensen's functional equation, J. Inequal. Pure Appl. Math. 4, no. 1, Art. ID 4 (2003).
  5. L. Cadariu and V. Radu, On the stability of the Cauchy functional equation: a fixed point approach, Grazer Math. Ber. 346 (2004), 43-52.
  6. L. Cadariu and V. Radu, Fixed point methods for the generalized stability of functional equations in a single variable, Fixed Point Theory and Applications 2008, Art. ID 749392 (2008).
  7. S.O. Carlsson, Orthogonality in normed linear spaces, Ark. Mat. 4 (1962),297-318. https://doi.org/10.1007/BF02591506
  8. P.W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. 27 (1984), 76-86. https://doi.org/10.1007/BF02192660
  9. S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg 62 (1992), 59-64. https://doi.org/10.1007/BF02941618
  10. S. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific Publishing Company, New Jersey, London, Singapore and Hong Kong, 2002.
  11. S. Czerwik, Stability of Functional Equations of Ulam-Hyers-Rassias Type, Hadronic Press, Palm Harbor, Florida, 2003.
  12. 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
  13. J. Diaz and B. Margolis, A fixed point theorem of the alternative for contractions on a generalized complete metric space, Bull. Amer. Math. Soc. 74 (1968), 305-309. https://doi.org/10.1090/S0002-9904-1968-11933-0
  14. C.R. Diminnie, A new orthogonality relation for normed linear spaces, Math. Nachr. 114 (1983), 197-203. https://doi.org/10.1002/mana.19831140115
  15. F. Drljevic, On a functional which is quadratic on A-orthogonal vectors, Publ. Inst. Math. (Beograd) 54 (1986), 63-71.
  16. M. Fochi, Functional equations in A-orthogonal vectors, Aequationes Math. 38 (1989), 28-40. https://doi.org/10.1007/BF01839491
  17. R. Ger and J. Sikorska, Stability of the orthogonal additivity, Bull. Polish Acad. Sci. Math. 43 (1995), 143-151.
  18. 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
  19. K. Hensel, Ubereine news Begrundung der Theorie der algebraischen Zahlen, Jahresber. Deutsch. Math.-Verein. 6 (1897), 83-88.
  20. D.H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. USA 27 (1941), 222-224. https://doi.org/10.1073/pnas.27.4.222
  21. D.H. Hyers, G. Isac and Th.M. Rassias, Stability of Functional Equations in Several Variables, Birkhauser, Basel, 1998.
  22. G. Isac and Th.M. Rassias, Stability of ${\varphi}$-additive mappings: Appications to nonlinear analysis, Internat. J. Math. Math. Sci. 19 (1996), 219-228. https://doi.org/10.1155/S0161171296000324
  23. R.C. James, Orthogonality in normed linear spaces, Duke Math. J. 12 (1945), 291-302. https://doi.org/10.1215/S0012-7094-45-01223-3
  24. R.C. James, Orthogonality and linear functionals in normed linear spaces, Trans. Amer. Math. Soc. 61 (1947), 265-292. https://doi.org/10.1090/S0002-9947-1947-0021241-4
  25. S. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press, Palm Harbor, Florida, 2001.
  26. Y. Jung and I. Chang, The stability of a cubic type functional equation with the fixed point alternative, J. Math. Anal. Appl. 306 (2005), 752-760. https://doi.org/10.1016/j.jmaa.2004.10.017
  27. 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
  28. A. Khrennikov, Non-Archimedean analysis: quantum paradoxes, dynamical systems and biological models, Mathematics and its Applications 427, Kluwer Academic Publishers, Dordrecht, 1997.
  29. D. Mihet and V. Radu, On the stability of the additive Cauchy functional equation in random normed spaces, J. Math. Anal. Appl. 343 (2008), 567-572. https://doi.org/10.1016/j.jmaa.2008.01.100
  30. M. Mirzavaziri and M.S. Moslehian, A fixed point approach to stability of a quadratic equation, Bull. Braz. Math. Soc. 37 (2006), 361-376. https://doi.org/10.1007/s00574-006-0016-z
  31. 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
  32. M.S. Moslehian, On the stability of the orthogonal Pexiderized Cauchy equation, J. Math. Anal. Appl. 318, (2006), 211-223. https://doi.org/10.1016/j.jmaa.2005.05.052
  33. M.S. Moslehian and Th.M. Rassias, Orthogonal stability of additive type equations, Aequationes Math. 73 (2007), 249-259. https://doi.org/10.1007/s00010-006-2868-0
  34. M.S. Moslehian and Gh. Sadeghi, A Mazur-Ulam theorem in non-Archimedean normed spaces, Nonlinear Anal.-TMA 69 (2008), 3405-3408. https://doi.org/10.1016/j.na.2007.09.023
  35. 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
  36. L. Paganoni and J. Ratz, Conditional function equations and orthogonal additivity, Aequationes Math. 50 (1995), 135-142. https://doi.org/10.1007/BF01831116
  37. C. Park, Fixed points and Hyers-Ulam-Rassias stability of Cauchy-Jensen functional equations in Banach algebras, Fixed Point Theory and Applications 2007, Art. ID 50175 (2007).
  38. C. Park, Generalized Hyers-Ulam-Rassias stability of quadratic functional equations: a fixed point approach, Fixed Point Theory and Applications 2008, Art. ID 493751 (2008).
  39. C. Park and J. Park, Generalized Hyers-Ulam stability of an Euler-Lagrange type additive mapping, J. Difference Equat. Appl. 12 (2006), 1277-1288. https://doi.org/10.1080/10236190600986925
  40. A.G. Pinsker, Sur une fonctionnelle dans l'espace de Hilbert, C. R. (Dokl.) Acad. Sci. URSS, n. Ser. 20 (1938), 411-414.
  41. V. Radu, The fixed point alternative and the stability of functional equations, Fixed Point Theory 4 (2003), 91-96.
  42. Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297-300. https://doi.org/10.1090/S0002-9939-1978-0507327-1
  43. Th.M. Rassias, On the stability of the quadratic functional equation and its applications, Studia Univ. Babes-Bolyai Math. 43 (1998), 89-124.
  44. Th.M. Rassias, The problem of S.M. Ulam for approximately multiplicative mappings, J. Math. Anal. Appl. 246 (2000), 352-378. https://doi.org/10.1006/jmaa.2000.6788
  45. Th.M. Rassias, On the stability of functional equations in Banach spaces, J. Math. Anal. Appl. 251 (2000), 264-284. https://doi.org/10.1006/jmaa.2000.7046
  46. Th.M. Rassias (ed.), Functional Equations, Inequalities and Applications, Kluwer Academic Publishers, Dordrecht, Boston and London, 2003.
  47. J. Ratz, On orthogonally additive mappings, Aequationes Math. 28 (1985), 35-49. https://doi.org/10.1007/BF02189390
  48. J. Ratz and Gy. Szabo, On orthogonally additive mappings IV, Aequationes Math. 38 (1989), 73-85. https://doi.org/10.1007/BF01839496
  49. F. Skof, Proprieta locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano 53 (1983), 113-129. https://doi.org/10.1007/BF02924890
  50. 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
  51. Gy. Szabo, Sesquilinear-orthogonally quadratic mappings, Aequationes Math. 40 (1990), 190-200. https://doi.org/10.1007/BF02112295
  52. S.M. Ulam, Problems in Modern Mathematics, Wiley, New York, 1960.
  53. F. Vajzovic, Uber 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) (1967), 73-81.

Cited by

  1. Effects of Perception of Job Characteristics on Innovation Behavior and Innovation Resistance vol.11, pp.4, 2013, https://doi.org/10.21871/kjfm.2020.12.11.4.7