On mean nonexpansive mappings and the Lifshitz constant
A mapping \(T\) of a metric space \(X\) into itself is called Lipschitzian if there exists a positive number \(k\) such that \(d(Tx,Ty) \leq k d(x,y)\) for \(x, y \in X.\) The smallest such \(k\) is called the Lipschitz constant \(k(T)\) of \(T\). If \(\alpha = (\alpha_1, \dots,\alpha_n\)), where the entries are positive real numbers adding up to 1, the mapping \(T\) is called an \(\alpha\)-nonexpansive mapping if, for each \(x,y \in M\), \(\{\Sigma \alpha_j d(T^j(x), T^j(y)) \leq d(x,y): j = 1, \dots,n\}\). The map \(T\) is called mean expansive when the multi-index \(\alpha\) is not specified. With these definitions, it is shown how the theory of polynomials can be used to describe the asymptotic behavior of the sequence of Lipschitz constants of the iterates of mean nonexpansive mappings. As consequences, using the Lifshitz theorem, some fixed point theorems are proved. Further, it is shown that the fixed point set of every mean nonexpansive mapping on a strictly convex Banach space is closed and convex.
scientific article; zbMATH DE number 6096039