Working from the comments in CL's answer:
You're talking about doing partial matches. e.g. Mich returns Michael, Walt returns Walter, Mary returns Mary Sue, etc.
EDIT:
Using a little reverse LIKE trickery and some input padding, you get this:
SELECT *
FROM People_Table
WHERE first_name || ' ' last_name LIKE '%' || (SELECT replace(inputString, ' ', '%')) || '%'
OR last_name || ' ' first_name LIKE '%' || (SELECT replace(inputString, ' ', '%')) || '%'
For double names like "Mary Sue Jones", the following will all match:
"Mary Jones", "Jones", "Sue Jones", "Jones Sue", "Sue Mary", "Mary Sue Jones", etc.
The drawback to this method is that if the portions of a double name are out of order in the search string, it will not match. You'll have to go outside this SQLite predicate to deal with that.