# ON STABILITY PROBLEMS WITH SHADOWING PROPERTY AND ITS APPLICATION

• Chu, Hahng-Yun (Department of Mathematics Chungnam National University) ;
• Han, Gil-Jun (Department of Mathematics Education Dankook University) ;
• Kang, Dong-Seung (Department of Mathematics Education Dankook University)
• Received : 2009.06.02
• Published : 2011.07.31

#### Abstract

Let $n{\geq}2$ be an even integer. We investigate that if an odd mapping f : X ${\rightarrow}$ Y satisfies the following equation $2_{n-2}C_{\frac{n}{2}-1}rf$\sum\limits^n_{j=1}{\frac{x_j}{r}}$\;+\;{\sum\limits_{i_k{\in}\{0,1\} \atop {{\sum}^n_{k=1}\;i_k={\frac{n}{2}}}}\;rf$\sum\limits^n_{i=1}(-1)^{i_k}{\frac{x_i}{r}}$=2_{n-2}C_{{\frac{n}{2}}-1}\sum\limits^n_{i=1}f(x_i),$ then f : X ${\rightarrow}$ Y is additive, where $r{\in}R$. We also prove the stability in normed group by using shadowing property and the Hyers-Ulam stability of the functional equation in Banach spaces and in Banach modules over unital C-algebras. As an application, we show that every almost linear bijection h : A ${\rightarrow}$ B of unital $C^*$-algebras A and B is a $C^*$-algebra isomorphism when $h(\frac{2^s}{r^s}uy)=h(\frac{2^s}{r^s}u)h(y)$ for all unitaries u ${\in}$ A, all y ${\in}$ A, and s = 0, 1, 2,....

#### References

1. J.-H. Bae and W.-G. Park, On the generalized Hyers-Ulam-Rassias stability in Banach modules over a $C^\ast$-algebra, J. Math. Anal. Appl. 294 (2004), no. 1, 196-205. https://doi.org/10.1016/j.jmaa.2004.02.009
2. C. Baak, D.-H. Boo, Th. M. Rassias, Generalized additive mapping in Banach modules and isomorphisms between $C^\ast$-algebras, J. Math. Anal. Appl. 314 (2006), no. 1, 150-161. https://doi.org/10.1016/j.jmaa.2005.03.099
3. H. Y. Chu and D. S. Kang, On the stability of an n-dimensional cubic functional equation, J. Math. Anal. Appl. 325 (2007), no. 1, 595-607. https://doi.org/10.1016/j.jmaa.2006.02.003
4. G.-L. Forti, Hyers-Ulam stability of functional equations in several variables, Aequationes Math. 50 (1995), no. 1-2, 143-190. https://doi.org/10.1007/BF01831117
5. G.-L. Forti, Comments on the core of the direct method for proving Hyers-Ulam stability of functional equations, J. Math. Anal. Appl. 295 (2004), no. 1, 127-133. https://doi.org/10.1016/j.jmaa.2004.03.011
6. 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
7. D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 222-224. https://doi.org/10.1073/pnas.27.4.222
8. R. V. Kadison and G. Pedersen, Means and convex combinations of unitary operators, Math. Scand. 57 (1985), no. 2, 249-266. https://doi.org/10.7146/math.scand.a-12116
9. R. V. Kadison and J. R. Ringrose, Fundamentals of the Theory of Operator Algebras. Vol. I, Elementary theory. Pure and Applied Mathematics, 100. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983.
10. S.-H. Lee, H. Koh, and S.-H. Ku, Investigation of the stability via shadowing property, J. Inequal. Appl. 2009 (2009), Art. ID 156167, 12 pp.
11. 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
12. F. Skof, Local properties and approximation of operators, Rend. Sem. Mat. Fis. Milano 53 (1983), 113-129. https://doi.org/10.1007/BF02924890
13. J. Tabor, Locally expanding mappings and hyperbolicity, Topol. Methods Nonlinear Anal. 30 (2007), no. 2, 335-343.
14. J. Tabor and J. Tabor, General stability of functional equations of linear type, J. Math. Anal. Appl. 328 (2007), no. 1, 192-200. https://doi.org/10.1016/j.jmaa.2006.05.022
15. S. M. Ulam, Problems in Morden Mathematics, Wiley, New York, 1960.