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) $.