You can't send messages using a Facebook application. You used to be able to do that, but the (predictable?) colossal amount of abuse led to the revocation of this ability.
Provided Alice, your user, has given you the necessary extended permissions, you have the following options:
- Post to Alice's wall on her behalf
- Send email to Alice
- Create events on behalf of Alice
- invite Bob (not your user) to said events
- Issue a request/invitation on behalf of Alice to Bob
- Issue a request from the App to Alice