existence of solutions using Gronwall

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

Quote:

Let

such that

,

,

where

are continuous on

. Show that the solutions of the IVP

,

exist on

.

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

Quote:

Suppose that

is a continuous solution of

,

,

where

are continuous and

for

. Then

,

.

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

Quote:

Let

,

,

,

, and

is bounded. Then every solution of

,

, exists on

.

Any help would be much appreciated!