Honam Mathematical Journal (호남수학학술지)

The Honam Mathematical Society (호남수학회)
권호 표지
DOI QR코드

DOI QR Code

ON THE STABILITY OF THE PEXIDER EQUATION IN SCHWARTZ DISTRIBUTIONS VIA HEAT KERNEL

  • Received : 2011.08.17
  • Accepted : 2011.10.11
  • Published : 2011.12.25
Download PDF
⟨ Previous Next ⟩

Abstract

We consider the Hyers-Ulam-Rassias stability problem $${\parallel}u{\circ}A-{\upsilon}{\circ}P_1-w{\circ}P_2{\parallel}{\leq}{\varepsilon}({\mid}x{\mid}^p+{\mid}y{\mid}^p)$$ for the Schwartz distributions u, ${\upsilon}$, w, which is a distributional version of the Pexider generalization of the Hyers-Ulam-Rassias stability problem ${\mid}(x+y)-g(x)-h(y){\mid}{\leq}{\varepsilon}({\mid}x{\mid}^p+{\mid}y{\mid}^p)$, x, $y{\in}\mathbb{R}^n$, for the functions f, g, h : $\mathbb{R}^n{\rightarrow}\mathbb{C}$.

Keywords

References

  1. J. A. Baker, Distributional methods for functional equations, Aeq. Math. 62 (2001), 136-142. https://doi.org/10.1007/PL00000134
  2. P. W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. 27(1984), 76-86. https://doi.org/10.1007/BF02192660
  3. J. Chung, A distributional version of functional equations and their stabilities, Nonlinear Analysis 62(2005), 1037-1051. https://doi.org/10.1016/j.na.2005.04.016
  4. J. Chung, Hyers-Ulam-Rassias stability of Cauchy equation in the space of Schwartz distributions, J. Math. Anal. Appl. 300(2004), 343-350. https://doi.org/10.1016/j.jmaa.2004.06.022
  5. J. Chung, Stability of functional equations in the space of distributions and hyperfunctions, J. Math. Anal. Appl. 286 (2003), 177-186. https://doi.org/10.1016/S0022-247X(03)00468-2
  6. 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), 107-114. https://doi.org/10.1016/j.jmaa.2004.03.009
  7. 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), 23-25.
  8. 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
  9. Z. Gajda, On stability of additive mappings, Internat. J. Math. Math. Sci. 14(1991), 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), 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. Y. H. Lee and K.W. Jun, A generalization of the Hyers-Ulam-Rassias stability of the Pexider equation, J. Math. Anal. Appl. 246(2000), 627-638. https://doi.org/10.1006/jmaa.2000.6832
  14. T. Matsuzawa, A calculus approach to hyperfunctions III, Nagoya Math. J. 118(1990), 133-153. https://doi.org/10.1017/S0027763000003032
  15. 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
  16. Th. M. Rassias, On the stability of linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72(1978), 297-300. https://doi.org/10.1090/S0002-9939-1978-0507327-1
  17. L. Schwartz, Theorie des Distributions, Hermann, Paris, 1966.
  18. F. Skof, Proprieta locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano 53(1983), 113-129. https://doi.org/10.1007/BF02924890
  19. S. M. Ulam, Problems in Modern Mathematics, Chapter VI, Wiley, New York, 1964.