DOI QR코드

DOI QR Code

ORTHOGONALITY AND LINEAR MAPPINGS IN BANACH MODULES

YUN, SUNGSIK

  • Received : 2015.09.03
  • Accepted : 2015.09.05
  • Published : 2015.11.30

Abstract

Using the fixed point method, we prove the Hyers-Ulam stability of lin- ear mappings in Banach modules over a unital C*-algebra and in non-Archimedean Banach modules over a unital C*-algebra associated with the orthogonally Cauchy- Jensen additive functional equation.

Keywords

Hyers-Ulam stability;orthogonally Cauchy-Jensen additive functional equation;fixed point;non-Archimedean Banach module over C*-algebra, orthogonality space

References

  1. 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
  2. ______: Stability of Functional Equations of Ulam-Hyers-Rassias Type. Hadronic Press, Palm Harbor, Florida, 2003.
  3. ______: Functional Equations and Inequalities in Several Variables. World Scientific Publishing Company, New Jersey, London, Singapore and Hong Kong, 2002.
  4. 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
  5. P.W. Cholewa: Remarks on the stability of functional equations. Aequationes Math. 27 (1984), 76-86. https://doi.org/10.1007/BF02192660
  6. S.O. Carlsson: Orthogonality in normed linear spaces. Ark. Mat. 4 (1962), 297-318. https://doi.org/10.1007/BF02591506
  7. ______: Fixed point methods for the generalized stability of functional equations in a single variable. Fixed Point Theory and Applications 2008, Art. ID 749392 (2008).
  8. ______: On the stability of the Cauchy functional equation: a fixed point approach. Grazer Math. Ber. 346 (2004), 43-52.
  9. L. Cădariu & V. Radu: Fixed points and the stability of Jensen's functional equation, J. Inequal. Pure Appl. Math. 4, no. 1, Art. ID 4 (2003).
  10. G. Birkhoff: Orthogonality in linear metric spaces. Duke Math. J. 1 (1935), 169-172. https://doi.org/10.1215/S0012-7094-35-00115-6
  11. ______: Orthogonality in normed linear spaces: a survey II. Relations between main orthogonalities. Extracta Math. 4 (1989), 121-131.
  12. J. Alonso & C. Benítez: Orthogonality in normed linear spaces: a survey I. Main properties. Extracta Math. 3 (1988), 1-15.
  13. ______: 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
  14. 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
  15. G. Isac & Th.M. Rassias: Stability of ψ-additive mappings: Appications to nonlinear analysis. Internat. J. Math. Math. Sci. 19 (1996), 219-228. https://doi.org/10.1155/S0161171296000324
  16. D.H. Hyers, G. Isac & Th.M. Rassias: Stability of Functional Equations in Several Variables. Birkhäuser, Basel, 1998.
  17. D.H. Hyers: On the stability of the linear functional equation. Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 222-224. https://doi.org/10.1073/pnas.27.4.222
  18. K. Hensel: Ubereine news Begrundung der Theorie der algebraischen Zahlen. Jahresber. Deutsch. Math. Verein 6 (1897), 83-88.
  19. S. Gudder & 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
  20. R. Ger & J. Sikorska: Stability of the orthogonal additivity. Bull. Polish Acad. Sci. Math. 43 (1995), 143-151.
  21. M. Fochi: Functional equations in A-orthogonal vectors. Aequationes Math. 38 (1989), 28-40. https://doi.org/10.1007/BF01839491
  22. F. Drljević: On a functional which is quadratic on A-orthogonal vectors. Publ. Inst. Math.(Beograd) 54 (1986), 63-71.
  23. C.R. Diminnie: A new orthogonality relation for normed linear spaces. Math. Nachr. 114 (1983), 197-203. https://doi.org/10.1002/mana.19831140115
  24. J. Diaz & 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
  25. 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
  26. M.S. Moslehian & 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
  27. R.V. Kadison & J.R. Ringrose: Fundamentals of the Theory of Operator Algebras. Academic Press, New York, 1983.
  28. S. Jung: Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis. Hadronic Press, Palm Harbor, Florida, 2001.
  29. M.S. Moslehian & Th.M. Rassias: Orthogonal stability of additive type equations. Aequationes Math. 73 (2007), 249-259. https://doi.org/10.1007/s00010-006-2868-0
  30. ______: 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
  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. Mirzavaziri & 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
  33. D. Miheţ & 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
  34. A. Khrennikov: Non-Archimedean analysis: quantum paradoxes, dynamical systems and biological models. Mathematics and its Applications 427, Kluwer Academic Publishers, Dordrecht, 1997.
  35. A.K. Katsaras & 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
  36. Th.M. Rassias (ed.): Functional Equations, Inequalities and Applications. Kluwer Academic Publishers, Dordrecht, Boston and London, 2003.
  37. ______: 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
  38. ______: 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
  39. Th.M. Rassias, On the stability of the quadratic functional equation and its applications, Studia Univ. Babeş-Bolyai Math. 43 (1998), 89-124.
  40. 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
  41. V. Radu: The fixed point alternative and the stability of functional equations. Fixed Point Theory 4 (2003), 91-96.
  42. A.G. Pinsker: Sur une fonctionnelle dans l’espace de Hilbert. C. R. (Dokl.) Acad. Sci. URSS, n. Ser. 20 (1938), 411-414.
  43. C. Park & 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
  44. ______: Generalized Hyers-Ulam-Rassias stability of quadratic functional equations: a fixed point approach. Fixed Point Theory and Applications 2008, Art. ID 493751(2008).
  45. C. Park: Fixed points and Hyers-Ulam-Rassias stability of Cauchy-Jensen functional equations in Banach algebras. Fixed Point Theory and Applications 2007, Art. ID50175 (2007).
  46. L. Paganoni & J. Rätz: Conditional function equations and orthogonal additivity. Aequationes Math. 50 (1995), 135-142. https://doi.org/10.1007/BF01831116
  47. F. Vajzović: Über das Funktional H mit der Eigenschaft: (x, y) = 0 ⇒ H(x + y) +H(x - y) = 2H(x) + 2H(y). Glasnik Mat. Ser. III 2 (1967), no. 22, 73-81.
  48. S.M. Ulam: Problems in Modern Mathematics. Wiley, New York, 1960.
  49. Gy. Szabó: Sesquilinear-orthogonally quadratic mappings. Aequationes Math. 40 (1990), 190-200. https://doi.org/10.1007/BF02112295
  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. F. Skof: Proprietà locali e approssimazione di operatori. Rend. Sem. Mat. Fis. Milano 53 (1983), 113-129. https://doi.org/10.1007/BF02924890
  52. J. Rätz & Gy. Szabó: On orthogonally additive mappings IV . Aequationes Math. 38(1989), 73-85. https://doi.org/10.1007/BF01839496
  53. J. Rätz: On orthogonally additive mappings. Aequationes Math. 28 (1985), 35-49. https://doi.org/10.1007/BF02189390