Multiplicative type functional equations arising from characterization problems (Q443933)

The main results obtained in this paper are the following: 1. If \(f_x, f_y, f_U, f_V: (0,1)\to\mathbb{R}\) are nonnegative measurable functions satisfying the equation \[ f_U(x)\cdot f_V(y)= f_x\Biggl({1- y\over 1- xy}\Biggr)\cdot f_x(1- xy)\cdot{y\over 1-xy} \] for almost all \((x,y)\in (0,1)^2\) such that they are positive on some sets of positive Lebesgue measure, then there exist positive constants \(p\), \(q\), \(r\), \(\varepsilon_i\), \(i= 1,2,3,4\), with \(\varepsilon_1\cdot \varepsilon_4= \varepsilon_2\cdot \varepsilon_3\) such that \[ f_x(x)= \varepsilon_1\cdot x^{p-1}(1- x)^{q-1},\;x\in(0,1)\text{ a.e.},\quad f_y(y)= \varepsilon_2\cdot y^{p+q-1}(1- y)^{r-1},\;y\in (0,1)\text{ a.e.}, \] \[ f_U(u)= \varepsilon_3\cdot u^{r-1}(1- u)^{q-1},\;u\in (0,1)\text{ a.e.},\quad f_V(v)= \varepsilon_4\cdot v^{q+r-1}(1- v)^{p-1},\;v\in (0,1)\text{ a.e.}. \] 2. If \(f,g,p,q: \mathbb{R}_+\to \mathbb{R}\) are nonnegative measurable functions which are positive on some sets of positive Lebesgue measure and which satisfy the equation \[ f(x)\cdot g(y)= p(x+ y)\cdot q\Biggl({x\over q}\Biggr),\quad (x,y)\in \mathbb{R}^2_+, \] then \[ f(x)= A\cdot \exp[ax+ b\ln x]\text{ a.a. }x\in\mathbb{R}_+,\quad g(x)= B\cdot\exp[ax+ (c- b)\ln x]\text{ a.a. }x\in\mathbb{R}_+, \] \[ p(x)= C\cdot\exp[ax+ c\ln ]\text{ a.e. }x\in\mathbb{R}_+,\quad q(x)= D\cdot\exp[b\ln x- c\ln(x+ 1)]\text{ a.a. }x\in R_+, \] where \(a,b,c\in\mathbb{R}\) and \(A,B,C,D\in\mathbb{R}\) are arbitrary constants with \(AB= CD\).