Pl post OS details. Are you restoring the 10.2.0.3 backup to the 10.2.0.3 HOME ? Your error message indicates that you are trying to open the upgraded 11gR2 database using the 10.2.0.3 HOME.
ORA-00201: control file version 188.8.131.52.0 incompatible with ORACLE version 10.2.0.3.0
If you installed 11gR2 binaries in a separate folder/home, it should not overwrite the binaries in the 10.2.0.3 HOME.
It has replaced the binaries, I have checked properly. I have renamed the sqlplus file in o_h/bin for 11g and when I went to o_h/bin in 10g the sqlplus is renamed.I tried to move the entire folder but it also not worked.
The os is AIX, 5L 64 BIT
Then I can only assume that software install instructions were not followed, or were executed incorrectly.