In addition to the problems addressed by others, I think there may be problems with the file name comparison regex in the OP. (Corion's reply Re: Array question also addresses this by using a different regex.)

The regex  $string_A =~ /^$string_B/i is anchored only at the beginning of the string, so if  $string_A eq 'foobar' and  $string_B eq 'foo' the two strings will match. (The strings being compared in the OP seem to consist of a file name and another file name with date appended, so this behavior may be correct.) A complete (case insensitive) match could be achieved by matching with  /^$string_B$/i using the  $ anchor.

Second, and perhaps more important, a string interpolation like  /^$string_B/ causes any regex metacharacters like  . ^ $ ( ) [ ] that are present in the string to be active in the compiled regex. (The most pesky metacharacter in this case is probably  . (dot): match any* character.) The usual practice would be to metaquote the intepolated string in some way, e.g.,  /^\Q$string_B\E/i or with the quotemeta function prior to interpolation.

* Well, any except a newline unless the  //s 'dot matches all' regex modifier is used. See s in the Modifiers section of perlre.