Few reason can cause this problem:
1) Battery, try fix battery in other similar phone, to ensure the battery performance.
2) On/Off Switch, try feel the switch, a broken on/off switch could cause tis problem too, its the cheapest among all possibilities.
3) Check the battery pin, poor or not touching connection pin also will result in can't on. Open the battery cover, push battery towards the connection pin to make tight connection, try tis, it might work!


Good Luck !!!!