Renewal equations on the semi-axis

We consider the renewal equation
$$ \varphi(x)=g(x)+\int_0^x\varphi(x-t)\,dF(t), \qquad g\in L_1(0;\infty), $$
where $F$ is the distribution function of a non-negative random variable. If $F$ has a non-trivial absolutely continuous component or is a distribution of absolutely continuous type, then we prove that the solution of the renewal equation can be written as follows:
$$ \varphi=\varphi_1+\varphi_2+\biggl[\int_0^{\infty}x\,dF(x)\biggr]^{-1}\int_0^{\infty}g(x)\,dt, $$
where $\varphi_1\in L_1(0;\infty)$, $\varphi_2\in C[0;\infty)$, and $\varphi_2(+\infty)=0$
If $g$ is bounded and $g(+\infty)=0$, then $\varphi_1(+\infty)=0$.
The proof is based on the structural factorization of the renewal equation into absolutely continuous, discrete, and singular components.