So what you want is a regular expression with a backreference.

my @array = qw( foo bar baz quux xyzzy);
my @matched;
for my $elem ( @array ) {
    push @matched, $elem if $elem =~ m/(.)\1/;
}
print "$_\n" for @matched;
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See perlre and search for "backreference" or for "find first doubled" for more explanation.