Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

ON THE HYERS-ULAM-RASSIAS STABILITY OF THE JENSEN EQUATION IN DISTRIBUTIONS

  • Lee, Eun-Gu (Department of Internet Business Dongyang Mirae University) ;
  • Chung, Jae-Young (Department of Mathematics Kunsan National University)
  • Received : 2010.04.02
  • Accepted : 2010.08.25
  • Published : 2011.04.30
Download PDF
⟨ Previous Next ⟩

Abstract

We consider the Hyers-Ulam-Rassias stability problem ${\parallel}2u{\circ}\frac{A}{2}-u{\circ}P_1-u{\circ}P_2{\parallel}{\leq}{\varepsilon}({\mid}x{\mid}^p+{\mid}y{\mid}^p)$, $x,y{\in}{\mathbb{R}}^n$ for the Schwartz distributions u, which is a distributional version of the Hyers-Ulam-Rassias stability problem of the Jensen functional equation ${\mid}2f(\frac{x+y}{2})-f(x)-F(y){\mid}{\leq}{\varepsilon}({\mid}x{\mid}^p+{\mid}y{\mid}^p)$, $x,y{\in}{\mathbb{R}}^n$ for the function f : ${\mathbb{R}}^n{\rightarrow}{\mathbb{C}}$.

Keywords

References

  1. J.-H. Bae, D.-O. Lee, and W. G. Park, On the stability of the Jensen's equation in Banach modules, J. Appl. Math. Comput. 11 (2003), no. 1-2, 423-430.
  2. J. A. Baker, Distributional methods for functional equations, Aequationes Math. 62 (2001), no. 1-2, 136-142. https://doi.org/10.1007/PL00000134
  3. P. W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. 27 (1984), no. 1-2, 76-86. https://doi.org/10.1007/BF02192660
  4. J. Chung, A distributional version of functional equations and their stabilities, Nonlinear Anal. 62 (2005), no. 6, 1037-1051. https://doi.org/10.1016/j.na.2005.04.016
  5. J. Chung, Hyers-Ulam-Rassias stability of Cauchy equation in the space of Schwartz distributions, J. Math. Anal. Appl. 300 (2004), no. 2, 343-350.
  6. J. Chung, Stability of functional equations in the space of distributions and hyperfunctions, J. Math. Anal. Appl. 286 (2003), no. 1, 177-186.
  7. J. Chung, S.-Y. Chung, and D. Kim, The stability of Cauchy equations in the space of Schwartz distributions, J. Math. Anal. Appl. 295 (2004), no. 1, 107-114. https://doi.org/10.1016/j.jmaa.2004.03.009
  8. J. Chung, S.-Y. Chung, and D. Kim, Une caracterisation de l'espace de Schwartz, C. R. Acad. Sci. Paris Ser. I Math. 316 (1993), no. 1, 23-25.
  9. Z. Gajda, On stability of additive mappings, Internat. J. Math. Math. Sci. 14 (1991), no. 3, 431-434. https://doi.org/10.1155/S016117129100056X
  10. P. Gavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), no. 3, 431-436. https://doi.org/10.1006/jmaa.1994.1211
  11. L. Hormander, The Analysis of Linear Partial Differential Operator I, Springer-Verlag, Berlin-New York, 1983.
  12. D. H. Hyers, On the stability of the linear functional equations, Proc. Nat. Acad. Sci. USA 27 (1941), 222-224. https://doi.org/10.1073/pnas.27.4.222
  13. S. M. Jung, Hyers-Ulam-Rassias stability of Jensen's equations and its application, Proc. Amer. Math. Soc. 126 (1998), no. 11, 3137-3143. https://doi.org/10.1090/S0002-9939-98-04680-2
  14. Z. Kominek, On a local stability of the Jensen functional equation, Demonstratio Math. 22 (1989), no. 2, 499-507.
  15. T. Matsuzawa, A calculus approach to hyperfunctions III, Nagoya Math. J. 118 (1990), 133-153. https://doi.org/10.1017/S0027763000003032
  16. C.-G. Park, Generalized Jensen's equations in a Banach module, Southeast Asian Bull. Math. 29 (2005), no. 6, 1117-1123.
  17. C.-G. Park, Universal Jensen's equations in Banach modules over a $C^{\*}$-algebra and its unitary group, Acta Math. Sin. (Engl. Ser.) 20 (2004), no. 6, 1047-1056. https://doi.org/10.1007/s10114-004-0409-0
  18. C.-G. Park, Generalized Jensen's equations in Banach modules over a unital $C^{*}$-algebra, Southwest J. Pure Appl. Math. 2002 (2002), no. 2, 52-63.
  19. C.-G. Park and W.-G. Park, On the stability of the Jensen's equation in a Hilbert module, Bull. Korean Math. Soc. 40 (2003), no. 1, 53-61. https://doi.org/10.4134/BKMS.2003.40.1.053
  20. C.-G. Park and W.-G. Park, On the Jensen's equation in Banach modules, Taiwanese J. Math. 6 (2002), no. 4, 523-531. https://doi.org/10.11650/twjm/1500407476
  21. Th. M. Rassias, On the stability of functional equations in Banach spaces, J. Math. Anal. Appl. 251 (2000), no. 1, 264-284. https://doi.org/10.1006/jmaa.2000.7046
  22. Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), no. 2, 297-300. https://doi.org/10.1090/S0002-9939-1978-0507327-1
  23. L. Schwartz, Theorie des Distributions, Hermann, Paris, 1966.
  24. S. M. Ulam, Problems in Modern Mathematics, Chapter VI, Wiley, New York, 1964.