I am trying (for fun!) to study the ongoing COVID-19 pandemic and have the following question. we know that an exponential function satisfies the following functional equation: \begin{equation} \frac{f(x+T)}{f(x)} = e^T = \text{constant} \end{equation} for some period $T$. For instance the function $f(x)=2^x$ satisfies \begin{equation} \frac{f(x+1)}{f(x)} = 2 \end{equation} Now, since here in Italy the growth rate of the infected is falling, the function describing the total amount of covid-19 cases since the start of the outbreak might satisfy the following equation: \begin{equation} \frac{f(x+T)}{f(x)} = g(x)\qquad \text{with $g$ decreasing to 1} \end{equation}
My question is this: how to solve this, is the solution existing, is it unique? Any hint on how I could approach this?
Let $f(x)$ be given for $x\in[0,T)$, and let $g(x)$ be given for $x\in[0,b]$, for some $T>0$ and $b\geq T$. Then the equation $$ f(x+T)=g(x)f(x) , $$ has a unique solution for $x\in[0,b]$. In other words, the solution $f$ is unique as a function defined on $[0,T+b]$.