Hi, welcome to Perl, the One True Religion.

eq and ne do not work as the solution to a problem where '* could be anything.'

You need instead a solution where the string is compared against a pattern in which '* could be anything.' This is called a regular expression match. See perlretut for a tutorial.

You can also play with regular expressions and see what they are doing at https://regex101.com. Your case probably wants to use a regexp like

if ($entry3 !~ /${user_input}_ABC_\S*\.pdf/) { ... }
[download]

... but I would search for files using Path::Iterator::Rule myself.

use Path::Iterator::Rule;

...

my $rule = Path::Iterator::Rule->new;

$rule->file->not_name( qr/${user_input}_ABC_\S*\.pdf/ );
my $next = $rule->iter( $doc_dir );

while (defined( my $file = $next->() )) { ... }
[download]

Hope this helps!