When a nonlinear operator has no fixed point?

Question

Let B be a normed vector space and $f: B \to B$ be a nonlinear operator. It is well known that the Banach fixed-point theorem or the Brouwer fixed-point theorem gives conditions which imply that a operator $f$ has a fixed point. My question is that is there a result on nonexistence of fixed points of a nonlinear operator. That is conditions which ensure that the operator $f$ has no fixed point.