Browse > Article
http://dx.doi.org/10.4134/BKMS.2007.44.4.851

APPROXIMATION OF CAUCHY ADDITIVE MAPPINGS  

Roh, Jai-Ok (DEPARTMENT OF MATHEMATICS HALLYM UNIVERSITY)
Shin, Hui-Joung (DEPARTMENT OF MATHEMATICS CHUNGNAM NATIONAL UNIVERSITY)
Publication Information
Bulletin of the Korean Mathematical Society / v.44, no.4, 2007 , pp. 851-860 More about this Journal
Abstract
In this paper, we prove that a function satisfying the following inequality $${\parallel}f(x)+2f(y)+2f(z){\parallel}{\leq}{\parallel}2f(\frac{x}{2}+y+z){\parallel}+{\epsilon}({\parallel}x{\parallel}^r{\cdot}{\parallel}y{\parallel}^r{\cdot}{\parallel}z{\parallel}^r)$$ for all x, y, z ${\in}$ X and for $\epsilon{\geq}0$, is Cauchy additive. Moreover, we will investigate for the stability in Banach spaces.
Keywords
Hyers-Ulam stability; Cauchy additive mapping; Jordan-von Neumann type Cauchy Jensen functional equation;
Citations & Related Records

Times Cited By Web Of Science : 2  (Related Records In Web of Science)
Times Cited By SCOPUS : 2
연도 인용수 순위
1 J. M. Rassias, On approximation of approximately linear mappings by linear mappings, J. Funct. Anal. 46 (1982), no. 1, 126-130   DOI
2 Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), no. 2, 297-300   DOI   ScienceOn
3 S. M. Ulam, A collection of mathematical problems, Interscience Tracts in Pure and Applied Mathematics, no. 8 Interscience Publishers, New York-London 1960
4 Th. M. Rassias and P. Semrl, On the behavior of mappings which do not satisfy Hyers- Ulam stability, Proc. Amer. Math. Soc. 114 (1992), no. 4, 989-993   DOI   ScienceOn
5 D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U. S. A. 27 (1941), 222-224   DOI   ScienceOn
6 K. Jun and Y. Lee, A generalization of the Hyers-Ulam-Rassias stability of the Pexider- ized quadratic equations, J. Math. Anal. Appl. 297 (2004), no. 1, 70-86   DOI   ScienceOn
7 C. Park, Homomorphisms between Poisson JC*-algebras, Bull. Braz. Math. Soc. (N.S.) 36 (2005), no. 1, 79-97   DOI
8 C. Park, Hyers-Ulam-Rassias stability of homomorphisms in quasi-Banach algebras, Bull. Sci. Math. (to appear)
9 C. Park, Y. Cho, and M. Han, Functional inequalities associated with Jordan-von Neumann-type additive functional equations, J. Inequal. Appl. (2007), 1-13   DOI
10 Z. Gajda, On stability of additive mappings, Internat. J. Math. Math. Sci. 14 (1991), no. 3, 431-434   DOI   ScienceOn
11 D. H. Hyers, G. Isac, and Th. M. Rassias, Stability of functional equations in several vari- ables, Progress in Nonlinear Differential Equations and their Applications, 34. Birkhauser Boston, Inc., Boston, MA, 1998
12 S. M. Jung, Hyers-Ulam-Rassias stability of functional equations in mathematical anal- ysis, Hadronic Press, Inc., Palm Harbor, FL, 2001