Convergence of a recursion in \$ L ^ 1 (0,1) \$

Leave $$mu_0> 0$$, $$a> 0$$, $$b> 0$$Y $$f (t)$$, $$g (t)> 0$$, $$p (t)$$ be some continuously differentiable functions on $$mathbb {R}$$.

I am looking for several tools to study the stability of the following recursive formula
$$h_n (t) = frac {- mu_ {n-1} f & # 39; (t)} {g ( mu_ {n-1} f (t))}, ; forall t in [0,1],$$

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

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