Suppose a function $t \mapsto x(t)$ is exponentially decreasing if and only if $x(t) \le c_1 e^{-\lambda_1 t} x(0)$ for some positive constants $c_1$ and $\lambda_1$. Suppose now we have another function $t \mapsto y(t)$. So what is the possible weakest relation between $x(t)$ and $y(t)$ such that $y$ is also exponentially decreasing, that is, there exists some positive constants $c_2$ and $\lambda_2$ such that $y(t) \le c_2 e^{-\lambda_2 t} y(0)$? Maybe this problem is ill-posed, but what kind of mathematical concepts I can use to describe a ''weak'' relation between two functions such that they share some properties such as exponentially decreasing?
Of course, one of the sufficient conditions is that $y(t) \le x(t)$ for all $t$, but I believe there are much weaker conditions.
If the functions are continuous, $\limsup_{t \to ∞} y(t)/x(t) < ∞$ should be enough. This is also denoted by $y ∈ O(x)$ (big O notation).