On functional equations of the multiplicative type

This paper aims to determine the general solution f:F2→Sf:F2→S of the equation f(ϕ(x,y,u,v))=f(x,y)f(u,v) for suitable conditions on the function ϕ:F4→F2ϕ:F4→F2, where FF will denote either RR or CC, and S   is a multiplicative semigroup. Using this result, we determine the general solution of several functional equations studied earlier, namely f(ux+vy,uy+vx)=f(x,y)f(u,v); f(ux+(λ−1)vy,vx+uy+(λ−2)vy)=f(x,y)f(u,v); f(ux−vy,uy−vx)=f(x,y)f(u,v); f(ux+vy,uy−vx)=f(x,y)f(u,v); f(ux+vy,uy−vx)=f(x,y)f(v,u); and f(ux−vy,uy+v(x+y))=f(x,y)f(u,v).