Let $\mu_0>0$, $a>0$, $b>0$, and $f(t)$, $g(t)>0$, $p(t)$ be some continuously differentiable functions over $\mathbb{R}$.

I am looking for various tools to study the stability of the following recursive formula $$h_n(t)=\frac{-\mu_{n-1} f'(t)}{g(\mu_{n-1} f(t))}, \; \forall t\in [0,1],$$

$$\mu_n=\frac{1}{a}\int_0^1h_n(t)\int_{t}^1p\left(b - \int_{0}^{s}h_n(r)dr\right)dsdt.$$

I specifically want to know if $h_n(t)$ is convergent in $L^1(0,1)$.