Korean Journal of Mathematics

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

DOI QR Code

APPROXIMATE ADDITIVE MAPPINGS IN 2-BANACH SPACES AND RELATED TOPICS: REVISITED

  • YUN, SUNGSIK (Department of Financial Mathematics Hanshin University)
  • Received : 2015.09.03
  • Accepted : 2015.09.09
  • Published : 2015.09.30
Download PDF
⟨ Previous Next ⟩

Abstract

W. Park [J. Math. Anal. Appl. 376 (2011) 193-202] proved the Hyers-Ulam stability of the Cauchy functional equation, the Jensen functional equation and the quadratic functional equation in 2-Banach spaces. But there are serious problems in the control functions given in all theorems of the paper. In this paper, we correct the statements of these results and prove the corrected theorems. Moreover, we prove the superstability of the Cauchy functional equation, the Jensen functional equation and the quadratic functional equation in 2-Banach spaces under the original given conditions.

Keywords

References

  1. S. M. Ulam, A Collection of the Mathematical Problems, Interscience Publ., New York, 1960.
  2. D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. 27 (1941) 222-224. https://doi.org/10.1073/pnas.27.4.222
  3. S. Gahler, 2-metrische Raume und ihre topologische Struktur, Math. Nachr. 26 (1963) 115-148. https://doi.org/10.1002/mana.19630260109
  4. S. Gahler, Lineare 2-normierte Raumen, Math. Nachr. 28 (1964) 1-43. https://doi.org/10.1002/mana.19640280102
  5. W.-G. Park, Approximate additive mappings in 2-Banach spaces and related topics, J. Math. Anal. Appl. 376 (2011) 193-202. https://doi.org/10.1016/j.jmaa.2010.10.004
  6. S. Gahler, Uber 2-Banach-Raume, Math. Nachr. 42 (1969) 335-347. https://doi.org/10.1002/mana.19690420414
  7. A. White, 2-Banach spaces, Doctorial Diss., St. Louis Univ., 1968.
  8. A. White, 2-Banach spaces, Math. Nachr. 42 (1969) 43-60. https://doi.org/10.1002/mana.19690420104