General Solution of Some Functional Equations Related to the Determinant of Some Symmetric Matrices

Abstract

Abstract. In this paper, we determine the general solution of the functional equation f (ux+vy, uy+vx) = g(x, y) h(u, v) where f,g,h: M 2 -» R are unknown functions. We also treat the equation f (ux + vy, uy + vx, zw) = g(x,y,z) h(u,v,w) where /, g,h : R 3 -» R are unknown functions. Our method is elementary and we do not use any regularity conditions. 1991 Mathematics Subject Classification: Primary 39B22. Introduction Let us define

Solutions of some functional equations related to the determinant of certain matrices

“…We prove the theorem by induction. The properties are true for n = 1 (see [3]) and n = 2 (see section 2 of this paper). Suppose that Properties (P 4) and (P 5) hold for n and let us prove them for n + 1.…”

On functional equations of the multiplicative type

“…Recently, Akkouchi and Rhali in [2] expanded the work of Chung and Sahoo [6]. They examined a functional equation that was related to the determinant of an n × n matrix.…”

“…Let K be the real or complex field and λ ∈ K * = K \ {0}. In [2], Akkouchi and Rhali, using a variation of the method proposed in [6], proved the following theorem:…”

“…The question that arises is whether f (x, y) = x 2 − y 2 is the only solution or there are other solutions. Chung and Sahoo in[6] have shown that there are other solutions besides f (x, y) = x 2 − y 2 . Chung and Sahoo in[6] proved the following theorem:Theorem 1.2.…”

On two functional equations and their solutions

“…The interested reader should refer to [1][2][3][4][6][7][8] for an in-depth account on the subject of functional equations.…”