cecompanies, could be your battery. Even new batteries sometimes fail. Here is my answer to a similar question " I would replace the battery first, next check the battery contacts with a multimeter and the USB cord plugged in. See what voltage you get on the contacts. It is easily possible that you have some pins corroded, bend, or broken. Also check for debris in the dock connector. If your dock connector checks out okay, check the fuses and see what you get. Check the image posted for fuse location (left is 3G right is 3GS.) Dimensions are 3GS fuse is 2mm x 1.20mm and the 3G one is 1mm x 0.30mm approximately.) Based on these specs the appropriate part number (Littelfuse Inc) seems to be 0435002 for 3G and 0467002 for 3GS. A small note about 3GS: the above fuse size is 0603, a little smaller of the original. The correct size is 0805 (2012 Metric), but unfortunately Littelfuse doesn't have this size available. An alternative could be MFU08052 from Vishay/Beyschlag Inc."
The fuse is actually a 0.050 ohm 1/6W 1% resistor in a 0402 package.
Hope this helps, good Luck