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Journal of Inequalities and Applications
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  • Research Article
  • Open Access

Inequalities in Additive -isometries on Linear -normed Banach Spaces

Journal of Inequalities and Applications20062007:070597

https://doi.org/10.1155/2007/70597

©  C. Park and T. M. Rassias. 2007

  • Received: 5 December 2005
  • Accepted: 17 October 2006
  • Published:

Abstract

We prove the generalized Hyers-Ulam stability of additive -isometries on linear -normed Banach spaces.

Keywords

  • Banach Space
  • Normed Banach Space

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Authors’ Affiliations

(1)
Department of Mathematics, Hanyang University, Seoul, 133-791, South Korea
(2)
Department of Mathematics, National Technical University of Athens, Zografou Campus, Athens, 15780, Greece

References

  1. Aleksandrov AD: Mappings of families of sets. Soviet Mathematics Doklady 1970, 11: 116–120.MATHGoogle Scholar
  2. Baker JA: Isometries in normed spaces. The American Mathematical Monthly 1971,78(6):655–658. 10.2307/2316577MathSciNetView ArticleMATHGoogle Scholar
  3. Bourgain J: Real isomorphic complex Banach spaces need not be complex isomorphic. Proceedings of the American Mathematical Society 1986,96(2):221–226. 10.1090/S0002-9939-1986-0818448-2MathSciNetView ArticleMATHGoogle Scholar
  4. Chu H-Y, Park C, Park W-G: The Aleksandrov problem in linear 2-normed spaces. Journal of Mathematical Analysis and Applications 2004,289(2):666–672. 10.1016/j.jmaa.2003.09.009MathSciNetView ArticleMATHGoogle Scholar
  5. Dolinar G: Generalized stability of isometries. Journal of Mathematical Analysis and Applications 2000,242(1):39–56. 10.1006/jmaa.1999.6649MathSciNetView ArticleMATHGoogle Scholar
  6. Gevirtz J: Stability of isometries on Banach spaces. Proceedings of the American Mathematical Society 1983,89(4):633–636. 10.1090/S0002-9939-1983-0718987-6MathSciNetView ArticleMATHGoogle Scholar
  7. Gruber PM: Stability of isometries. Transactions of the American Mathematical Society 1978, 245: 263–277.MathSciNetView ArticleMATHGoogle Scholar
  8. Ma Y: The Aleksandrov problem for unit distance preserving mapping. Acta Mathematica Scientia 2000,20(3):359–364.MathSciNetMATHGoogle Scholar
  9. Mazur S, Ulam S: Sur les transformation isometriques d'espaces vectoriels normes. Comptes Rendus de l'Académie des Sciences 1932, 194: 946–948.MATHGoogle Scholar
  10. Mielnik B, Rassias TM: On the Aleksandrov problem of conservative distances. Proceedings of the American Mathematical Society 1992,116(4):1115–1118. 10.1090/S0002-9939-1992-1101989-3MathSciNetView ArticleMATHGoogle Scholar
  11. Rassias TM: Properties of isometric mappings. Journal of Mathematical Analysis and Applications 1999,235(1):108–121. 10.1006/jmaa.1999.6363MathSciNetView ArticleMATHGoogle Scholar
  12. Rassias TM: On the A. D. Aleksandrov problem of conservative distances and the Mazur-Ulam theorem. Nonlinear Analysis. Theory, Methods & Applications 2001,47(4):2597–2608. 10.1016/S0362-546X(01)00381-9MathSciNetView ArticleMATHGoogle Scholar
  13. Rassias TM, Xiang S: On mappings with conservative distances and the Mazur-Ulam theorem. Univerzitet u Beogradu. Publikacije Elektrotehničkog Fakulteta. Serija Matematika 2000, 11: 1–8 (2001).MathSciNetMATHGoogle Scholar
  14. Xiang S: Mappings of conservative distances and the Mazur-Ulam theorem. Journal of Mathematical Analysis and Applications 2001,254(1):262–274. 10.1006/jmaa.2000.7276MathSciNetView ArticleMATHGoogle Scholar
  15. Rassias TM, Šemrl P: On the Mazur-Ulam theorem and the Aleksandrov problem for unit distance preserving mappings. Proceedings of the American Mathematical Society 1993,118(3):919–925. 10.1090/S0002-9939-1993-1111437-6MathSciNetView ArticleMATHGoogle Scholar
  16. Cho YJ, Lin PCS, Kim SS, Misiak A: Theory of 2-Inner Product Spaces. Nova Science, New York; 2001:xii+330.MATHGoogle Scholar
  17. C. Park and T. M. Rassias, On a generalized Trif ’s mapping in Banach modules over a C*-algebra, Journal of the Korean Mathematical Society 43 (2006), no. 2, 323–356.MathSciNetView ArticleMATHGoogle Scholar
  18. Chu H-Y, Lee K, Park C: On the Aleksandrov problem in linear-normed spaces. Nonlinear Analysis. Theory, Methods & Applications 2004,59(7):1001–1011.MathSciNetMATHGoogle Scholar
  19. Rassias TM: On the stability of the linear mapping in Banach spaces. Proceedings of the American Mathematical Society 1978,72(2):297–300. 10.1090/S0002-9939-1978-0507327-1MathSciNetView ArticleMATHGoogle Scholar
  20. Jun K-W, Bae J-H, Lee Y-H: On the Hyers-Ulam-Rassias stability of an-dimensional Pexiderized quadratic equation. Mathematical Inequalities & Applications 2004,7(1):63–77.MathSciNetView ArticleMATHGoogle Scholar
  21. Jun K-W, Lee Y-H: On the Hyers-Ulam-Rassias stability of a Pexiderized quadratic inequality. Mathematical Inequalities & Applications 2001,4(1):93–118.MathSciNetView ArticleMATHGoogle Scholar
  22. Jung S-M: Hyers-Ulam stability of Butler-Rassias functional equation. Journal of Inequalities and Applications 2005,2005(1):41–47. 10.1155/JIA.2005.41View ArticleMathSciNetMATHGoogle Scholar
  23. Miura T, Takahasi S-E, Hirasawa G: Hyers-Ulam-Rassias stability of Jordan homomorphisms on Banach algebras. Journal of Inequalities and Applications 2005,2005(4):435–441. 10.1155/JIA.2005.435MathSciNetView ArticleMATHGoogle Scholar
  24. Rassias TM: On the stability of functional equations in Banach spaces. Journal of Mathematical Analysis and Applications 2000,251(1):264–284. 10.1006/jmaa.2000.7046MathSciNetView ArticleMATHGoogle Scholar
  25. Rassias TM: The problem of S. M. Ulam for approximately multiplicative mappings. Journal of Mathematical Analysis and Applications 2000,246(2):352–378. 10.1006/jmaa.2000.6788MathSciNetView ArticleMATHGoogle Scholar
  26. Rassias TM, Šemrl P: On the Hyers-Ulam stability of linear mappings. Journal of Mathematical Analysis and Applications 1993,173(2):325–338. 10.1006/jmaa.1993.1070MathSciNetView ArticleMATHGoogle Scholar
  27. Trif T: On the stability of a functional equation deriving from an inequality of Popoviciu for convex functions. Journal of Mathematical Analysis and Applications 2002,272(2):604–616. 10.1016/S0022-247X(02)00181-6MathSciNetView ArticleMATHGoogle Scholar
  28. Park C, Rassias TM: On a generalized Trif's mapping in Banach modules over a-algebra. Journal of the Korean Mathematical Society 2006,43(2):323–356.MathSciNetView ArticleMATHGoogle Scholar

Copyright

© C. Park and T. M. Rassias. 2007

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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