On the explicit estimates for the power rate of convergence in the renewal theorem

The renewal equation $x(t)=y(t)+\int_0^t x(t-s)\,dF(s)$ is considered. The explicit estimates are obtained for
$$ \sup_t(1+\varepsilon t)^{\alpha}|x(t)-\lim_{t\to\infty}x(t)| $$
under the assumptions on the power decay of $y(t)$ and on the existence of moments of $F(t)$ for some classes of distribution functions $F(t)$. One of this classes include distributions having independent exponential component.