Very low battery life

Good chance it's done. Unused batteries boil over and corrode. I'd toss it away and get a new one, is t worth a fire.

yup, your batteries are dead, buy new ones.
Old ones are not holding a charge

The quality of the battery determines the life of the battery.
If the laptop battery is more than 3 - 4 years old then the battery could be worn out. Rechargeable batteries have a finite number of charge and discharge cycles and will lose their charge capacity over time, i.e. won't charge to 100% and gradually the charge reduces until the battery won't charge up at all. OR The battery shows a 100% charge but when the adapter is disconnected the battery drops off to zero capacity in a very short time. If the battery drops to an unacceptable charge level then the battery needs to be replaced. Rechargeable batteries will fail if stored in a discharged state for long periods.

If the rechargeable battery is more than 3 - 4 years old then the battery could be worn out. Rechargeable batteries have a finite number of charge and discharge cycles and will lose their charge capacity over time, i.e. won't charge to 100% and gradually the charge reduces until the battery won't charge up at all. OR The battery shows a 100% charge but when the adapter is disconnected the battery drops off to zero capacity in a very short time. If the battery drops to an unacceptable charge level then the battery needs to be replaced. Rechargeable batteries will fail if stored in a discharged state for long periods.

If unused for any length of time, rechargeable batteries lose their ability to retain a charge. Try it with a brand new, fully-charged battery. FWIW, I recharge all my camera and flashgun batteries monthly.

That sounds like your battery could be dead/not hold a charge. If you've charged your battery already and your camera still won't stay off and turns off right away, it's time to purchase a new battery for your Samsung TL105.

You can purchase replacement batteries quite cheap here:
http://www.amazon.com/Samsung-Digital-Camera-BP70A-Battery/dp/B003OEDQBE

If the troubleshooting section in the back of the user guide doesn't help, I sugget getting in touch with Sony and asking them if it's normal, the only thing I can suggest in the meantime, is taking the battery out while it's not in use, or recharge it a few hours before you're due to use it. Phones get like it after a few years and with technology advancing the way it is these days, you'd expect a battery life lasting longer than a week, not a day.

Things to consider:
1. When it's off, is it off or left on standby?
2. Where did you buy both the camera and batteries from?
3. When you charge it up, do you charge it up with a half charged battery or a totally dead battery?

Recharge your batteries for AT LEAST 15 hours, even if your charger's light indicates full charge. You should get at least 200 pics from a charge.

AA batteries are available in four basic varieties: Alkaline Nickel-Metal Hydride (NiMH) Nickel-Cadmium (NiCad) Photo Lithium (Li-FeS2) Alkaline and photo lithium are non-rechargeable, while NiMH and NiCad are rechargeable. Each has advantages and disadvantages. Non-Rechargeable vs. Rechargeable: Rechargeable batteries are desirable from an environmental standpoint because they are reusable. Self-discharge refers to the fact that batteries lose energy when unused and even when not in a camera or other device. Rechargeable batteries tend to have relatively high self-discharge rates, approximately 1-2% per day for nickel-based batteries. Non-rechargeable batteries generally have very long shelf lives and extremely slow self-discharge rates. This makes non-rechargeable batteries a better choice for infrequent usage. Non-rechargeable batteries are available fully charged in stores all over the world, which makes them a convenient choice for travelers or customers who have dead rechargeable batteries and no time to recharge. Photo Lithium Batteries (Li-FeS2) (non-rechargeable): Photo lithium batteries will yield the longest battery life of any AA battery, surpassing NiMH by 50-100% and surpassing alkaline by 100-500%, depending on the load. While they are more expensive than alkaline batteries, their additional energy capacity makes the cost the same or less per shot than alkaline batteries. Nickel-Metal Hydride Batteries (NiMH) (rechargeable): NiMH batteries are the lowest cost overall solution for users that take a lot of pictures (more than the equivalent of a roll of film per month) or use a lot of high-power features. The largest disadvantage to NiMH batteries is their fast self-discharge rate of 1-2% per day whether the batteries are in a camera or not. NOTE: NiMH batteries need to be completely charged and discharged a few times when new to achieve their full capacity. Rechargeable batteries will eventually fail. If you have been getting acceptable battery life and then see a decrease in life, either quickly or slowly over time, a worn-out battery may be the cause. Storing or charging the batteries in high temperature conditions will accelerate this potential failure. Alkaline Batteries (non-rechargeable): Although the cheapest and easiest to find, alkaline batteries yield the worst performance of all the chemistries in a digital camera. They lose capacity at high power drains and at low temperatures. Skiers and other winter outdoor enthusiasts may find them unsatisfactory. Alkaline batteries are frequently available in two types: High drain (ultra, titanium, maximum etc.) Regular The high drain versions are a premium product designed to operate better under heavy loads than the standard product. However, there is a trend of major brands to increase the performance of their standard battery to b