global $wpdb;
$query = "SELECT post_id FROM wp_postmeta WHERE meta_key='fake_author' AND meta_value='11111'";
$results = $wpdb->get_results($query);

foreach ($results as $result) {
    $query = "UPDATE wp_posts SET post_author='222222' WHERE ID='".$result->post_id."'";
    $wpdb->query($query);
}