see perldoc -f push and Tutorials (below the text explaining how to/when to post a new tut) or Super Search re iterating through arrays.

IOW, collect all the relevant $file_name entries into an array; at that point, iterating thru the individual files requires only that you attempt to inform yourself thru study of the references above.

Update: Of course (/me says belatedly), you'll need to consider executing your script from an appropriate directory or troubling yourself to learn about perldoc -f chdir, perldoc cwd, etc.