I consider the following equation
$$ m_t = \mu t + \int_{0}^{t} h(t-s)m_sds $$
Where $\mu>0$ is fixed and $h$ is a locally integrable and non negative function. I want to solve this equation.
I have a candidate given by
$$ m_t = \mu t + \mu\int_{0}^{t}H(t-s)sds $$
Where $H(t) = \sum_{n\geq}h^{*n}(t)$ with, for all $n\geq 2$
$$ h^{*n}(t) = \int_{0}^{t} h^{*n-1}(t-u)h(u)du $$
And $h^{*}(t) = h(t)$.
I get stuck at some point :
Indeed, trying to prove this directly, I get
$$ m_t = \mu t + \mu\int_{0}^{t}(h(t-s) + (H*h)(t-s))sds $$
We know that $h(t-s) = h^{*}(t-s)$ and by monotone convergence theorem
$$ (H*h)(t-s) = \int_{0}^{t}H(t-s-v)h(v)dv = \sum_{n\geq1} \int_{0}^{t}h^{*n}(t-s-v)h(v)dv = \sum_{n\geq1} h^{*n+1}(t-s) = H(t-s) $$
Thus we get
$$ m_t = \mu t + \mu\int_{0}^{t}(h(t-s) +H(t-s))sds $$
I doubt that the argument put in bold is right because I cannot see how to conclude using the last expression I wrote. Have you any idea on what could be wrong ? I would like to have some hint rather than answer please, except if all what I did is totally wrong of course.
Thank you a lot !