Conditional functional equations on restricted domains of measure zero

Abstract

Let R02=R2∖{(0,0)}, R⁎2={(x,y)∈R2:x2≠y2} and f:R02→R, g:R⁎2→R. In this paper we consider the Ulam–Hyers stability of the functional equationsf(ux+vy,uy−vx)=f(x,y)+f(u,v),f(ux−vy,uy+vx)=f(x,y)+f(u,v),g(ux−vy,uy−vx)=g(x,y)+g(u,v),g(ux+vy,uy+vx)=g(x,y)+g(u,v) for all (x,y,u,v)∈Γ, where Γ⊂R4 is of 4-dimensional Lebesgue measure zero. The above functional equations are modified versions of the equations in [9,11,14,18,24] which arise from number theory and are in connection with characterizations of determinant and permanent of two-by-two matrices.