if you have had your PSP for more than a year then it is PROBABLY the battery. Rechargable batteries lose their charge over time no matter what device it is. Never leave your device on the charger when fully charged, do not let the battery die completely before charging except for the first initial usage of the battery, and never let the battery be in hot enviroments. These few things can make a battery last longer but only about 8 months more. Batteries are not that expensive.
Depending on the Model Number of your PSP you can find batteries for about $20. You can even buy an extended battery for $20+. It may be thicker but it comes with a custom made backing and will give about double the charge time and overall life of the battery.