First, turn your phone off, wait for a couple minutes and on it again. If you find this worked then the error happened with the software. It can be resolved by doing the power-cycling.
Second, remove the battery and sim card if you have one, while the phone is still in a turn on position. Wait for a couple minutes and put the battery and sim card back. Restart the phone and check the result.
Also check on [http://www.sonyericsson.com/cws/support/phones/detailed/thephonewillnotturnonorthephoneswitchesitselfoff/w200i?cc=us&lc=en|Here] and here is more [http://www.sonyericsson.com/cws/support/phones/w200i?cc=us&lc=en|Support for your phone.] Good Luck and let us know what happened