This is a fairly common problem on some iPad models, ultimately caused by a battery that's no longer as fresh as it once was. The problem is this: Your iPad shuts itself off due to lack of charge so you plug it into your charger. Apple designed the iPad to turn itself on after its battery charge is greater than a certain point (I think it's 3%). The problem is, older batteries can register a certain amount of charge WITHOUT actually being charged. (If you've ever had your phone at, say, 10% and seen it immediately jump to 20% or more immediately after you plugged it in, you know what I mean.) So what happens is, when you plug in a charger the battery erroneously reports that it has >3% charge when it actually doesn't so the iPad turns itself on -- only to have to immediately shut off because the battery is completely dead. Then (because it's still plugged in) the battery again erroneously reports that it has >3% charge when it actually doesn't so the iPad turns itself on -- only to have to immediately shut off because the battery is completely dead. And repeat forever.
There are two practical solutions that I'm aware of, the first one I haven't had much success with. Plug in your charger and the moment you see the Apple logo boot screen hold down the POWER button. That will force it to shut off. Unfortunately there's no way to KEEP it off so the iPad will almost immediately boot up again. Repeat this cycle a LOT of times, like at least 50-60 times. Each cycle will build up a little bit of power in the battery. Eventually the battery will have JUST enough power to complete the boot cycle and get you to the unlock screen. Immediately slide-to-shut down the iPad, then plug it into your charger and allow it to charge normally. (Since you shut the iPad down properly it won't automatically restart again.)
The other solution that I've found to be more successful is to use a stronger charger. A lot of people mistakenly use the 5W iPhone charger, which won't work. iPads require a 10W or 12W charger, and if you look around you can find chargers that offer as much as 18.5W. The iPad Air has a 10W charger. If you can get a 12W or greater charger, plug it in and let it sit for awhile. It will continue doing its auto-reboot thing for awhile but eventually it will boot to the lock screen, at which point it will charge normally. (There's usually no reason to shut down the iPad again at this point.)
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