# existence of solutions using Gronwall

• Sep 20th 2010, 01:21 PM
hatsoff
existence of solutions using Gronwall
I must complete the following exercise by the end of tonight:

Quote:

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

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

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

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

exist on $(a,b)$.
It is suggested that we use the following formulation of the Gronwall inequality:

Quote:

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

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

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

$\displaystyle u(t)\leq f(t)+\int_{t_0}^t f(s)h(s)\exp\left(\int_s^t h(\sigma)d\sigma\right)ds$, $t\geq t_0$.
Also, I suspect (though I am not certain) that the following theorem, part of the unit of this exercise, might be useful:

Quote:

Let $D=(a,b)\times\mathbb{R}^n$, $-\infty\leq a, $t_0\in(a,b)$, $f\in C(D,\mathbb{R}^n)$, and $f$ is bounded. Then every solution of $x'=f(t,x)$, $x(t_0)=x_0$, exists on $(a,b)$.
Any help would be much appreciated!
• Sep 20th 2010, 01:46 PM
bkarpuz
Quote:

Originally Posted by hatsoff
I must complete the following exercise by the end of tonight:
...
Any help would be much appreciated!