iphone 5 on reboot mode and how to fix it

Try Some Troubleshooting?

1. Restart:  Press On/Off button until the Slide to Power Off slider appears, select Slide to Power Off and, after It shuts down, press the On/Off button until the Apple logo appears. Also, see Restart your iPhone, iPad, or iPod touch - Apple Support.
2. Reset:  Press the Home and On/Off buttons at the same time and hold them until the Apple logo appears (about 10-15 seconds). Also, see iPhone, iPad, iPod touch- Turning off and on (restarting) and resetting.
3. Restore:  Connect your iPhone to iTunes on your computer, backup, and then select Restore to Factory. Also, see Restore your iPhone, iPad, or iPod to factory settings and If you forgot the passcode for your iPhone, iPad, or iPod touch, or your device is disabled - Apple.

For help with the above on later models, including iPhone 7, 7s, 8 or later, iPad Pro 11" and iPad Pro 12.9" models, please refer to If your iPhone, iPad, or iPod touch won't turn on or is frozen.

If all else fails then make an appointment at the Apple Genius Bar for service. If you need to find an Apple Store - Find a Store - Apple. If you did not purchase it from Apple then you will need to look to the vendor for service or How to find your nearest Apple Authorized Service Provider (AASP) or Distributor (AAD).