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

• Journal title : Honam Mathematical Journal
• Volume 33, Issue 4,  2011, pp.467-485
• Publisher : The Honam Mathematical Society
• DOI : 10.5831/HMJ.2011.33.4.467
Title & Authors
ON THE STABILITY OF THE PEXIDER EQUATION IN SCHWARTZ DISTRIBUTIONS VIA HEAT KERNEL
Chung, Jae-Young; Chang, Jeong-Wook;

Abstract
We consider the Hyers-Ulam-Rassias stability problem $\small{{\parallel}u{\circ}A-{\upsilon}{\circ}P_1-w{\circ}P_2{\parallel}{\leq}{\varepsilon}({\mid}x{\mid}^p+{\mid}y{\mid}^p)}$ for the Schwartz distributions u, $\small{{\upsilon}}$, w, which is a distributional version of the Pexider generalization of the Hyers-Ulam-Rassias stability problem $\small{{\mid}(x+y)-g(x)-h(y){\mid}{\leq}{\varepsilon}({\mid}x{\mid}^p+{\mid}y{\mid}^p)}$, x, $\small{y{\in}\mathbb{R}^n}$, for the functions f, g, h : $\small{\mathbb{R}^n{\rightarrow}\mathbb{C}}$.
Keywords
Stability;Gauss transforms;heat kernel;distributions;tempered distribution;Cauchy equation;Pexider equation;
Language
English
Cited by
References
1.
J. A. Baker, Distributional methods for functional equations, Aeq. Math. 62 (2001), 136-142.

2.
P. W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. 27(1984), 76-86.

3.
J. Chung, A distributional version of functional equations and their stabilities, Nonlinear Analysis 62(2005), 1037-1051.

4.
J. Chung, Hyers-Ulam-Rassias stability of Cauchy equation in the space of Schwartz distributions, J. Math. Anal. Appl. 300(2004), 343-350.

5.
J. Chung, Stability of functional equations in the space of distributions and hyperfunctions, J. Math. Anal. Appl. 286 (2003), 177-186.

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.

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.

9.
Z. Gajda, On stability of additive mappings, Internat. J. Math. Math. Sci. 14(1991), 431-434.

10.
P. Gavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184(1994), 431-436.

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.

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.

14.
T. Matsuzawa, A calculus approach to hyperfunctions III, Nagoya Math. J. 118(1990), 133-153.

15.
Th. M. Rassias, On the stability of functional equations in Banach spaces, J. Math. Anal. Appl. 251(2000), 264-284.

16.
Th. M. Rassias, On the stability of linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72(1978), 297-300.

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.

19.
S. M. Ulam, Problems in Modern Mathematics, Chapter VI, Wiley, New York, 1964.