DearMHFmembers,

I need the proof of the following result but I don't know where can I get it.

Theorem.Let $\displaystyle A\in\mathrm{C}([0,\infty),\mathbb{R}_{n}^{n})$, $\displaystyle F\in\mathrm{C}([0,\infty),\mathbb{R}^{n})$ and $\displaystyle X_{0}\in\mathbb{R}^{n}$.

Then the unique solution $\displaystyle \varphi(\cdot,s)$ of the initial value problem

$\displaystyle \begin{cases}X^{\prime}=A(t)X+F,\ r\geq t\geq s\geq0\\ X(s)=X_{0}\end{cases}$,

where $\displaystyle r>0$ is fixed, is continuous in $\displaystyle s$.

I would be very glad if you can help me in this direction.

Thanks.

bkarpuz