Iphone 3Gs discharging

Hello i am using iphone 3Gs everything has been fine untill now because it discharges way faster than it supposed to. Sometimes i can not call it discharging because there is still 50% battery left and it just turns off and i have to wait with charger plugged in so it turns on again. Sometimes it's very fast if I take charger with myself and i plug it in very fast after it discharged it takes like 1 minute and iphone is on again with 50% battery left or so. Sometimes for example this time i am waiting for over 1 day and all i see is apple logo. This thing happend few months ago so i realised that when it is fully discharged its easier to turn it on because if apple logo on black screen is shown there is probably no chance to turn in on again unless wait till it is gone and it takes quiet a while. What i mean is iphone is not discharged but all the sympthoms are discharged iphone, actually it would be way better if it discharges for real so i can turn it on again fast, but now when there are half battery left i can't do that its annoying. Also my turn off/on button is broke now. Sorry for all the mistakes, and i am glad someone will help me