Finding an example

Question

I am looking for an example to have two distinct decreasing functions $g_1$ and $g_2$ between [0,1], such that their ratio ($\phi=\frac{g_1}{g_2}$) is a periodic function with period one, moreover for two distinct density function $f_1$ and $f_2$ the following equality holds, $f_1.g_1=f_2.g_2$.