I am interested in know, what is the correct practice when you change your position and have some unpublished research results (without any specific funding). Is it appropriate to use both affiliations? Can I use only the last one?
If you have unpublished research results (without any fundings), you should specify the place and your position at the time the work was conducted. Presently you may be affiliated to a new institute where you may get sufficient financial support for research, but this support was not available for you when you carried out previous work. So a mention of previous affiliation is a must. Funding body details are required at the time of submission in most of the journals, hence the information you provide must mot contradict with your present affiliation. Providing the information about both the affiliations will not only rule out any conflicts of interest but also boost your scientific reputation. And it is also one's moral duty to mention the place where we did the work.
You have to use the current place for affiliation. Normally there will be no provision for two affiliations. I would suggest that you should mention the fact that it was conducted at a different place in the acknowledgments.
Today I found the answer to your question as described by an Elsevier journal in its author guidelines. It also matches the previous answers given by various commenters. I would like to quote it below:
Present/permanent address. If an author has moved since the work described in the article was done, or was visiting at the time, a 'Present address' (or 'Permanent address') may be indicated as a footnote to that author's name. The address at which the author actually did the work must be retained as the main, affiliation address.
There is a way: If your former place was X and your new one is Y, and you wrote the paper at X, write your address as follows: "X, currently at Y". Then the situation becomes clear.
What about the author, if He is in International Cooperation Abroad for teaching at Universtiy2, while He continues to make researches connected and supported by its Country’s University1.
Nevertheless, The University2 forces him to write single affiliation (Univrsity2) even if the work is totally supported by its University1. If not his Contract is stopped! Is there an International rules or ethics?
See ethically you should add both the affiliations of your Univ 1 and 2 with your name as author. Else you can use the affiliation of Univ 2 in author's name and acknowledge thoroughly Univ 1 in the funding section of your paper.