# Refined Stability Results of Functional Equation in Four Variables

Kim, Hark-Mahn;Lee, Soon

• Accepted : 2013.07.18
• Published : 2015.03.23
• 25 6

#### Abstract

In this paper, we present the general solution of the functional equation $$rf(\frac{x+y+z+w}{s})+rf(\frac{x+y-z-w}{s})+rf(\frac{x-y+z-w}{s})+rf(\frac{x-y-z+w}{s})=tf(x)+tf(y)+tf(z)+tf(w)$$ and improve the Hyers-Ulam stability of the equation.

#### References

1. T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan, 2(1950), 64-66. https://doi.org/10.2969/jmsj/00210064
2. D. G. Bourgin, Classes of transformations and bordering transformations, Bull. Amer. Math. Soc., 57(1951), 223-237. https://doi.org/10.1090/S0002-9904-1951-09511-7
3. P. W. Cholewa, Remarks on the shability of functional equations, Aequationes Math., 27(1984), 76-86. https://doi.org/10.1007/BF02192660
4. 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
5. 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
6. D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. U.S.A., 27(1941), 222-224. https://doi.org/10.1073/pnas.27.4.222
7. S. Lee and K. Jun,Hyers-Ulam-Rassias stability of a quadratic type functional equation, Bull. Korean Math. Soc., 40(2003), 183-193. https://doi.org/10.4134/BKMS.2003.40.2.183
8. S. Lee and C. Park, Hyers-Ulam-Rassias stability of a functional equation in three variables, J. Chungcheong Math. Soc., 16(2)(2003), 11-21.
9. C. Park, Hyers-Ulam-Rassias stability of an even functional equation in four variables, Kyungpook Math J., 44(2)(2004), 299-304.
10. Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72(1978), 297-300. https://doi.org/10.1090/S0002-9939-1978-0507327-1
11. F. Skof, Proprieta locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano, 53(1983), 113-129. https://doi.org/10.1007/BF02924890
12. S. M. Ulam, A Collection of the Mathematical Problems, Interscience Publ. New York, 1960.

#### Acknowledgement

Supported by : National Research Foundation (NRF)