You will have to load jquery.formstyler first to use it in your ajax.js.
You can add it as another dependancie like jquery itself.
wp_enqueue_script( 'my-ajax-handle', plugin_dir_url( _FILE_ ) . 'guards_search/js/ajax.js', array( 'jquery', 'jquery.formstyler' ) );
Also you have a little syntax issue in:
wp_register_style('jquery.formstyler', plugin_dir_url( _FILE_ ) . 'guards_search/js/jquery.formstyler.min.js');
It should be register script, not style. same with the line below wp_enqueue_style('jquery.formstyler');.
add as wp_enqueue_script('jquery.formstyler'); script instead as style!