Plugin function inside custom plugin

You’re probably making the check too early. Make sure all plugins are loaded before trying to access the methods they provide. You can hook into plugins_loaded action to make sure all plugins are loaded before trying to use the functions they define.

class My_Plugin {
    public static function init() {
        if( function_exists( 'wp_bannerize' ) ) {
            echo "exist";
        } else {
            echo "not exist";
        } 
    }
    public function __construct() { ... }
}
add_action( 'plugins_loaded', array ( 'My_Plugin', 'init' ), 10 );