If you want to stay logged in to your membership account on your iPhone, please follow these steps:
On your iPhone, go to Settings > Safari. Scroll down to the "Privacy & Security" section. Then disable (turn from green to gray) Prevent Cross-site Tracking & Block All Cookies:
Once you do this, you should be able to stay logged in to your membership account for a long time. Also, make sure you check the "Remember Me" box: