Find $f(t)$ : $f(t) \leq \int_0^t f(s)ds$,$0 \leq t \leq 1$

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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