Denote \(T(t)\) the semigroup of linear operators defined on some Banach space \(B\). Asynchronous exponential growth occurs when one can determine a pair \((\lambda_0,v)\), \(\lambda_0\in{\mathbb{R}}\), \(0\neq v\in B\), such that \(e^{-\lambda_0 t}T(t)x - c v\to 0\) as \(t\to +\infty\) for every \(x\in B\) and some \(c\in{\mathbb{R}}\). The main result of the paper is an estimate of the rate of convergence of \(T(t)\) to an asynchronous equilibrium. The particular case of translation semigroups is examined.
The results obtained are applied to an equation in demography.