Let $\Big ( p_i(u) : i \in \mathbb{N}, u \in [0, \infty) \Big )$ be a sequence of functions between $0$ and $1$. Suppose that these functions satisfy, for some given non-negative coefficients $c_i \geq 0$, $i \in \mathbb{N}$, $$ \forall j \in \mathbb{N}, \quad p_{j}(t) = \int_{0}^t du \sum\limits_{i \in \mathbb{N}\\ i \neq j} c_i p_i(u) du. $$
Do I have enough information to deuduce that the sum of the integrand is absolute convergent?
It is clear to me that the sum in the integrand must be at least point-wise convergent, since the integral is finite (it must be equal to $p_{j}(t)$), hence the integrand must be finite. But what about uniformity? From a text I read it seems that uniformity should also follow from the expression...