A uniform asymptotic renewal theorem

Let $x(t)=x(t,y(\,\cdot\,),F(\,\cdot\,))$ (for probability distribution $F$ on $R_+$ and bounded function $y$) be the solution of the renewal equation
$$ x(t)=y(t)+\int_{[0,t)}x(t-s)F(ds). $$
Denote by $\mathfrak K$ a class of distributions $F$ such that each $F\in\mathfrak K$ has an absolutely continuous component $G$ with uniformly (over $\mathfrak K$) positive total mass and the corresponding class of densities $\frac{\partial G}{\partial t}$ is uniformly bounded on $R_+$ and relatively compact in $L_1 (R_+)$.
If nondecreasing function $\varphi$ on $R_+$ is such that $\varphi(t+s)\leqslant\varphi(t)\varphi(s)$, $\lim_{t\to\infty}\varphi(t+s)/\varphi(t)=1$, if $F\in\mathfrak K$ and the functions
$$ \int_{[t,\infty)}\varphi(s)F([s,\infty))\,ds,\quad\varphi(t)y(t),\quad\varphi(t)\int_{[t,\infty)}y(s)\,ds $$
converge uniformly to 0 as $t\to\infty$, then
$$ x(t)-\biggl(\int_{R_+}sF\,(ds)\biggr)^{-1}\int_{R_+}y(s)\,ds=o(1/\varphi(t)),\qquad t\to\infty, $$
uniformly in $F$ and $y$. The uniform exponential asymptotics of $x(t)$ is obtained also.