The problem is that your unlink isn't targeted at the FILE filehandle. It's trying to delete from whereever the script was run, except that it never gets there, because you're opening a directory with a file open command. Use "or die ("File open error:$!");" on the open and I bet it'll die.

You don't have to open files you are trying to delete. See unlink

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Update: Let's look at your script line by line:

open(FILE,"/cgi and perl");
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Problems:

1. You didn't test for a die condition.
2. You are opening a directory for no real reason
3. What you are trying for is done by opendir and readdir

   while(<FILE>){
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problems:

1. This will read FILE line by line (except that it's a directory), and put the line in $_ (which would never be used anyway)

       unlink(<*.txt>);
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This line alone should do what you want, if the script is run in the directory you want the text files deleted from.