Mean value theorem for vector valued function in $\mathcal{C}([0,a]\times E\times E, E)$ Let $a$ is a real number such that $a>0$, $E$ a Banach space and $f :[0,a]\times E\times E\rightarrow  E$ a continuous function.
Is the following statement correct?

For every $t\in [0,a]$, let $\overline{conv}$ the closure of the convex hull, then : $$\int_{0}^{t}f(s,u(s),v(s)) ds\in t\,.\overline{conv} \{f(s,u(s),v(s))\::s\in [0,t] )\}$$
  with $u,v\in \mathcal C([0,a],E)$

If so why this is correct?

In the end, you are simply integrating a continuous function. You always have 
$$
\int_0^a f(s)\,ds\in a\,\overline{\operatorname{conv}}\{f(s):\ s\in [0,a]\}.
$$
This is a straightforward consequence of the definition of a Riemman integral: the Riemann sums for your integral are of the form 
$$
\sum_j f(s_j)\, \Delta_j=a\,\sum_j f(s_j)\, \tfrac{\Delta_j} a,
$$
where $\sum_j\Delta_j=a$. The right-hand-side is a convex combination of values of $f$, times $a$. 
