Sony Ericsson Xperia X10 mini, that just only showed the Sony Ericsson logo and often restarted...
This is cause by an interrupted firmware, or software related problem issues.
1. Let the battery run flat
2. Download and install SEUS (Sony Ericsson Update Service) or other flasher like SEtool2
3. Connect it and flash it (It can easily be flashed as it charges with USB
It will take some time for the unit to start up normally after flash, like all other Sony Ericsson Handsets.