On mean nonexpansive mappings and the Lifshitz constant

We present how the theory of polynomials can be used to describe the asymptotic behaviour of the sequence of Lipschitz constants for iterates of mean nonexpansive mappings. We find, as consequences, using the Lifshitz theorem, some new fixed point theorems. We also prove that the fixed point set of every mean nonexpansive mapping is closed and convex, provided the Banach space is strictly convex.