foreach $a1(@a1){
  while(@a2 = <F2>)
  {
      ...
  }
}
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Note also that with the block structure quoted above, the  F2 filehandle in the nested while-loop will be "exhausted" after handling the first item in the the foreach-loop and will thereafter, I think, assign to the  @a2 array a list consisting of a single undef value. To be useful, the  F2 filehandle would need to be rewound after each pass through the while-loop; see seek. (bliako has already made this point here.)

A marginally better loop nesting structure would be

while (my $line2 = <F2>) {
    foreach my $line1 (@a1) {
        print $line1 if line2_appears_in_line1($line2, $line1);
        }
    }
[download]

Another point. If you want to find out if a line from file2 (always remember to chomp this line!) is exactly present within a line from file1, the comparison should be
    if ($line1 =~ /\Q$line2_chomped\E/) { ... }
or better still (because simpler and faster)
    if (index($line1, $line2_chomped) >= 0) { ... }
See index. (index is more appropriate here because no real regex matching seems needed, only an exact substring match.) Your code here seems to have this relationship wackbards.

A more advanced point. If file file2 is small enough, the technique described in haukex's Building Regex Alternations Dynamically article could be used to build a single regex that could be matched against each line of file file1 to determine which of these lines were to be printed. This approach would require only a single pass through each file, i.e., will be O(n), but will fail if file2 is much more than several hundred (or perhaps several thousand — YMMV) lines. This approach is capable of handling an unlimited number of lines in file1 however.

Update: Minor wording and spelling corrections.