Can U help with optimal search method?

Can you guys show the best way to find users by first and last name or by full name.
Also, when the user enters a name, I want to search for that name in both Cyrillic and Latin.
Any links, ideas?

Client could enter Name or LastName or together(Fullname (firstname lastname or lastname firstname) )
I used some techniques, but when I enter Zhandos Osipov it return Zhandos Zhandos.
Also, client could enter just part of the name like "Zhandos Osi" or "Osipov Zhandos"

In DB I have data like
Zhandos Osipov
Жандос Осипов
Жандос Жандос
Zhandos Zhandos
Shandos Osipov
Zhandos Osepov
Zhandos Erlan
Zhandos Omar
Zhandos Oskar