This is what I came up with: 
#!/bin/perl5

use strict;
use warnings;

@ARGV = qw( file1.txt file2.txt );

my $name;
my %hash;

while (<>){
  chomp;
  my $record = $_;
  if ( $record =~ /^>/ ){
    $name = $record;
    next;
  }
  else{
    $hash{$name} .= $record
  }
}

for my $key ( keys %hash ){
  print "$key\n$hash{$key}\n"
}

[download]
Updated: Use hash instead of a hashref

In reply to Re: how to merge the files of DNA sequences? by wfsp
in thread how to merge the files of DNA sequences? by newperler

Title:
Use:  <p> text here (a paragraph) </p>
and:  <code> code here </code>
to format your post, it's "PerlMonks-approved HTML":


  • Posts are HTML formatted. Put <p> </p> tags around your paragraphs. Put <code> </code> tags around your code and data!
  • Titles consisting of a single word are discouraged, and in most cases are disallowed outright.
  • Read Where should I post X? if you're not absolutely sure you're posting in the right place.
  • Please read these before you post! —
  • Posts may use any of the Perl Monks Approved HTML tags:
    a, abbr, b, big, blockquote, br, caption, center, col, colgroup, dd, del, details, div, dl, dt, em, font, h1, h2, h3, h4, h5, h6, hr, i, ins, li, ol, p, pre, readmore, small, span, spoiler, strike, strong, sub, summary, sup, table, tbody, td, tfoot, th, thead, tr, tt, u, ul, wbr
  • You may need to use entities for some characters, as follows. (Exception: Within code tags, you can put the characters literally.)
            For:     Use:
    & &amp;
    < &lt;
    > &gt;
    [ &#91;
    ] &#93;
  • Link using PerlMonks shortcuts! What shortcuts can I use for linking?
  • See Writeup Formatting Tips and other pages linked from there for more info.