A Functional Equation Related to Determinant of Some 3 x3 Symmetric Matrices and Its Pexiderized Form

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Charinthip Hengkrawit*
Wuttichai Suriyacharoen

Abstract

In this work, we present the general solution of a functional equation f (ux + vy,uyn+ zw)= (x,y,z)+ (u,v,w)+  (x,y,z) f (u,v,w) for all x,y,u,v,w, z gif.latex?\in gif.latex?\mathbb{R} , which arises from determinant of some symmetric 3 x 3 matrices. We also determine the general solution of its Pexiderized version f (ux+vy,uy + vx,zw)= gif.latex?g(x,y,z)+ h (u,v,w)+ gif.latex?l (x,y,z) n (u,v,w) for all x,y,u,v,w,z gif.latex?\ingif.latex?\mathbb{R},without any regularity assumptions on unknown functions f,gif.latex?g,h,gif.latex?l,n :gif.latex?\mathbb{R}3  gif.latex?\rightarrow gif.latex?\mathbb{R}.


Keywords: Determinant of matrix, functional equation, logarithmic function, multiplicative function


*Corresponding author:  Tel.: +662 564 4440 Fax: +662 564 4489


E-mail: charinthip@mathstat.sci.tu.ac.th


 

Article Details

Section
Original Research Articles

References

[1] Chung, J.K. and Sahoo, P.K., 2002. General solution of some functional equations related to the determinant of symmetric matrices. Demostratio Math., 35, 539-544.
[2] Houston, K.B. and Sahoo, P.K., 2008. On two functional equations and their solutions. Applied Mathematics Letters, 21, 974-977.