• 발행 : 2006.11.30

#### 초록

Let X, Y be vector spaces. It is shown that if an even mapping $f:X{\rightarrow}Y$ satisfies f(0) = 0 and $$(0.1)\;f(\frac {x+y} 2+z)+f(\frac {x+y} 2-z)+f(\frac {x-y} 2+z)+f(\frac {x-y} 2-z)=f(x)+f(y)+4f(z)$$ for all x, y, z ${\in}$X, then the mapping $f:X{\rightarrow}Y$ is quadratic. Furthermore, we prove the Cauchy-Rassias stability of the functional equation (0.1) in Banach spaces.

#### 참고문헌

1. C. Baak, S. Hong, and M. Kim, Generalized quadratic mappings of $\gamma$-type in several variables, J. Math. Anal. Appl. 310 (2005), 116-127 https://doi.org/10.1016/j.jmaa.2005.01.056
2. P. W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. 27 (1984), 76-86 https://doi.org/10.1007/BF02192660
3. 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
4. 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
5. D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 222-224
6. J. Kang, C. Lee and Y. Lee, A note on the Hyers-Ulam-Rassias stability of a quadratic equation, Bull. Korean Math. Soc. 41 (2004), 541-557 https://doi.org/10.4134/BKMS.2004.41.3.541
7. C. Park, On the Hyers-Ulam-Rassias stability of generalized quadratic mappings in Banach modules, J. Math. Anal. Appl. 291 (2004), 214-223 https://doi.org/10.1016/j.jmaa.2003.10.027
8. C. Park, Generalized quadratic mappings in several variables, Nonlinear Anal-ysis-TMA 57 (2004), 713-722 https://doi.org/10.1016/j.na.2004.03.013
9. C. Park, J. Park and J. Shin, Hyers-Ulam-Rassias stability of quadratic functional equations in Banach modules over a $C^{*}$-algebra, Chinese Ann. Math. Series B 24 (2003), 261-266 https://doi.org/10.1142/S0252959903000244
10. Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297-300
11. Th. M. Rassias, On the stability of the quadratic functional equation and its applications, Studia Univ. abes-Bolyai XLIII (1998), no. 3, 89-124
12. Th. M. Rassias, The problem of S.M. Ulam for approximately multiplicative mappings, J. Math. Anal. Appl. 246 (2000), 352-378 https://doi.org/10.1006/jmaa.2000.6788
13. 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
14. Th. M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Appl. Math. 62 (2000), 23-130 https://doi.org/10.1023/A:1006499223572
15. Th. M. Rassias and P. Semrl, On the Hyers-Ulam stability of linear mappings, J. Math. Anal. Appl. 173 (1993), 325-338 https://doi.org/10.1006/jmaa.1993.1070
16. Th. M. Rassias and K. Shibata, Variational problem of some quadratic functionals in complex analysis, J. Math. Anal. Appl. 228 (1998), 234-253 https://doi.org/10.1006/jmaa.1998.6129
17. F. Skof, Proprieta locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano 53 (1983), 113-129 https://doi.org/10.1007/BF02924890
18. T. Trif, Hyers-Ulam-Rassias stability of a quadratic functional equation, Bull. Korean Math. Soc. 40 (2003), 253-267 https://doi.org/10.4134/BKMS.2003.40.2.253
19. S. M. Ulam, Problems in Modern Mathematics, Wiley, New York, 1960

#### 피인용 문헌

1. Stability of a Bi-Additive Functional Equation in Banach Modules Over aC⋆-Algebra vol.2012, 2012, https://doi.org/10.1155/2012/835893
2. A FIXED POINT APPROACH TO THE CAUCHY-RASSIAS STABILITY OF GENERAL JENSEN TYPE QUADRATIC-QUADRATIC MAPPINGS vol.47, pp.5, 2010, https://doi.org/10.4134/BKMS.2010.47.5.987