### STABILITY OF TWO FUNCTIONAL EQUATIONS ARISING FROM DETERMINANT OF MATRICES

Choi, Chang-Kwon;Kim, Jongjin;Lee, Bogeun

• 투고 : 2015.09.11
• 발행 : 2016.07.31
• 8 1

#### 초록

Let $f:{\mathbb{R}}^3{\rightarrow}{\mathbb{R}}$. In this paper we prove the stability of functional inequalities ${\mid}f(ux+vy,uy-vx,zw)-f(x,y,z)f(u,v,w){\mid}{\leq}{\phi}(u,v,w)$ or ${\phi}(x,y,z)$, ${\mid}f(ux-vy,uy-vx,zw)-f(x,y,z)f(u,v,w){\mid}{\leq}{\phi}(u,v,w)$ or ${\phi}(x,y,z)$ for all $x,y,z,u,v,w{\in}{\mathbb{R}}$. Furthermore, we give refined descriptions of bounded functions satisfying the inequalities as in Albert and Baker [1].

#### 키워드

determinant;exponential functional equation;multiplicative function;stability

#### 참고문헌

1. M. Albert and J. A. Baker, Bounded solutions of a functional inequality, Canad. Math. Bull. 25 (1982), no. 4, 491-495. https://doi.org/10.4153/CMB-1982-071-9
2. J. A. Baker, J. Lawrence, and F. Zorzitto, The stability of the equation f(x + y) = f(x)f(y), Proc. Amer. Math. Soc. 74 (1979), no. 2, 242-246.
3. J. Chung, On an exponential functional inequality and its distributional version, Canad. Math. Bull. 58 (2015), no. 1, 30-43. https://doi.org/10.4153/CMB-2014-012-x
4. J. Chung and J. Chang, On two functional equations originating from number theory, Proc. Indian Acad. Sci. Math. Sci. 124 (2014), no. 4, 563-572. https://doi.org/10.1007/s12044-014-0200-9
5. J. Chung, T. Riedel, and P. K. Sahoo, Stability of functional equations arising from number theory and determinant of matrices, preprint.
6. J. K. Chung and P. K. Sahoo, General solution of some functional equations related to the determinant of some symmetric matrices, Demonstratio Math. 35 (2002), no. 3, 539-544.
7. D. H. Hyers, G. Isac, and Th. M. Rassias, Stability of functional equations in several variables, Birkhauser, 1998.
8. K. B. Houston and P. K. Sahoo, On two functional equations and their solutions, Appl. Math. Lett. 21 (2008), no. 9, 974-977. https://doi.org/10.1016/j.aml.2007.10.012
9. S. M. Jung and J. H. Bae, Some functional equations originating from number theory, Proc. Indian Acad. Sci. Math. Sci. 113 (2003), no. 2, 91-98. https://doi.org/10.1007/BF02829761