by Charalampos Filippatos
Last Updated October 26, 2017 11:20 AM - source

**Exercise :**

*Find all the continuous and non-negative functions $f(t)$ such that:*

$$f(t) \leq \int_0^t f(s)ds$$

**Attempt :**

I know there are some more "brutal" solving ways to this, such as a *Laplace* transformation or probably *Mean Value Theorem* tricks, but my idea is using the *Gronwall Inequality*, which says :

*Let $φ(t)$ be a continuous function in $[0,T]$. Suppose that $\exists k>0$ and another continuous function $f(t)$, such that :*

$$ φ(t) \leq f(t) + k\int_o^tφ(u)du, \forall t \in [0,T]$$

*Then : $$φ(t) \leq f(t) + k\int_0^t f(τ)e^{t-τ}dτ,\forall t \in [0,T]$$*

My problem is that I cannot see how to properly use the inequality or how to fix it, so I can show what is asked.

Any help would be appreciated !

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