foreach (@files)
{
   open FILE, "$_" or die $!;
      print "$_", br() x3;

      $_ = map m{([^/]+)\.txt}, $_;  #Strip out of the string the path and extension leaving only the filename intact
      ...
}

##</code><code>##

( my $name ) = m{ ([^/]+) \.suffix }x;

#  or,

my $name = ( m{ ([^/]+) \.suffix }x )[0];

#  or,

use File::Basename;
my $name = fileparse( $_ , qr{ \.suffix }x );