my $dirname= '/some/directory/with/files';
my $input= '2013-09-04';
opendir my $dh, $dirname
    or die "Couldn't read '$dirname': $!";
my @found= grep /^\Q$input\E/, readdir $dh;

...

##</code><code>##

use File::Glob qw( bsd_glob );
my @found= bsd_glob("$input*");