The problem is you are not writing the new string to the file. First you must seek to the start of the file, truncate it, then write the new string. My Mark.txt has 2 lines,NAME and ZANY, which gets changed to MAME ZAMY

The magic of processing @ARGV can also be used, as shown in the second example.

#!/usr/bin/perl

open (logfilefsv, '>>logfsv.txt');

$file = "Mark.txt";

open(IN, "+<", $file);
#my $old = "N";
#my $new = "M";
@lines = <IN>;

print logfilefsv @lines;

seek IN, 0, 0; # seek to top of file
truncate(IN, 0); # truncate old data

foreach $line (@lines) {
       $line =~ s/N/M/gi;
       print IN $line;
       }
__END__
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Or use Magic @ARGV processing, Mark.txt is shifted in from the command line

#!/usr/bin/perl

{
local ($^I, @ARGV) = ('.bak', shift );
 while (<>) {
    $_ =~ s/M/N/gi; 
    print "$_";
 }
}
__END__
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