Welcome to iPhone Forums, ngotseva!
There's a Google app, which will remember your account, even if you force close the app, as long as you don't log out.
Where exactly do you try to access your Gmail account on your iPhone?
Btw:
The other thread you started, discussing the same issue, has been removed. Besides fragmenting the conversation and making it hard to follow, duplicate posts and threads are against our rules.
Please keep any discussion of your problem to this thread.