A new method for constructing invariant subspaces

The new idea that is used in this article for producing non-trivial (closed) invariant subspaces of (bounded linear) operators on reflexive Banach spaces, is the use of fixed points of set-valued functions. The advantage of this new method is that it is reasonable to expect that the famous method of Lomonosov for producing invariant subspaces using fixed points of functions, can be viewed as a special case of the use of fixed points of set-valued functions. Further uses of this new idea and open questions are suggested at the end of the article.