Q: my iPhone has problems

Did you let it charge for at least 30 minutes? Do so using the wall charger and then reset it (hold down the home and sleep buttons, wait for the Apple logo, let go of the buttons) while it is still connected to power.

Make sure you have the latest version of iTunes on your computer, try various USB ports (do not use a hub) and disconnect any other non-essential USB devices. Restart your computer before you connect.

Do you have the latest version of iTunes?

Did you restart your computer?

Which USB port on your computer are you using?

Did you disconnect non-essential USB devices?

And another cable?

If you try another cable and that doesn't work I would guess the battery has failed on your iPhone and you will need to get it replaced.

In addition to the information that deggie has provided you, check out this support document. Get help with iOS update and restore errors - Apple Support

The information regarding error 2001 states:

Check your USB connections

Errors: 13, 14, 1600, 1601, 1602, 1603, 1604, 1611, 1643-1650, 2000, 2001, 2002, 2005, 2006, 2009, 4005, 4013, 4014. You might also see a message that includes "invalid response."

If the USB connection between your device and computer is interrupted, you might not be able to update or restore.

To narrow down the issue, try different equipment:

• Use the USB cable that came with your device, or a different Apple USB cable.
• Plug your cable into a different USB port directly on your computer. Don't plug it into your keyboard.
• Try a different computer.
• Solve other issues with the USB connection, then with your security software.

If you still see the error message, check for hardware issues

.

Check for hardware issues

Errors: 1, 3, 11, 12, 13, 14, 16, 20, 21, 23, 26, 27, 28, 29, 34, 35, 36, 37, 40, 53, 1002, 1004, 1011, 1012, 1014, 1667, or 1669.Your device or computer might have a hardware issue that stops the update or restore from completing.

1. Check that your security software and settings aren't blocking your device from communicating with the Apple update server.
2. Then try to restore your iOS device two more times while connected with a cable, computer, and network you know are good.
3. Make sure that your security software and settings are allowing communication between your device and update servers.
4. If you still see the error message when you update or restore, contact Apple support.