existence of solutions using Gronwall

I must complete the following exercise by the end of tonight:

Quote:

Let $\displaystyle f\in C((a,b)\times\mathbb{R}^n,\mathbb{R}^n)$ such that

$\displaystyle |f(t,x)|\leq p(t)|x|+q(t)$, $\displaystyle (t,x)\in(a,b)\times\mathbb{R}^n$,

where $\displaystyle p,q$ are continuous on $\displaystyle (a,b)$. Show that the solutions of the IVP

$\displaystyle x'=f(t,x)$, $\displaystyle x(t_0)=x_0$

exist on $\displaystyle (a,b)$.

It is suggested that we use the following formulation of the Gronwall inequality:

Quote:

Suppose that $\displaystyle u(t)$ is a continuous solution of

$\displaystyle \displaystyle u(t)\leq f(t)+\int_{t_0}^t h(s)u(s)ds$, $\displaystyle t\geq t_0$,

where $\displaystyle f,h$ are continuous and $\displaystyle h(t)\geq 0$ for $\displaystyle t\geq t_0$. Then

$\displaystyle \displaystyle u(t)\leq f(t)+\int_{t_0}^t f(s)h(s)\exp\left(\int_s^t h(\sigma)d\sigma\right)ds$, $\displaystyle t\geq t_0$.

Also, I suspect (though I am not certain) that the following theorem, part of the unit of this exercise, might be useful:

Quote:

Let $\displaystyle D=(a,b)\times\mathbb{R}^n$, $\displaystyle -\infty\leq a<b\leq\infty$, $\displaystyle t_0\in(a,b)$, $\displaystyle f\in C(D,\mathbb{R}^n)$, and $\displaystyle f$ is bounded. Then every solution of $\displaystyle x'=f(t,x)$, $\displaystyle x(t_0)=x_0$, exists on $\displaystyle (a,b)$.

Any help would be much appreciated!