Fixed point and nonlinear ergodic theorems for semigroups of nonlinear mappings. (Q2761640)

The authors present a survey of some recent results on the relation between the existence of an invariant subspace of \(\ell^{\infty}(S)\) -- the Banach space (endowed with the supremum norm) of bounded real functions on a semigroup -- and the existence of fixed points, or of ergodic properties of \(S\), in the case when \(S\) admits a representation as a semigroup of nonexpansive mappings acting on a Banach space.NEWLINENEWLINEThis chapter of the handbook is mainly concerned with the study of the nonlinear mean (called submean) of a subspace of \(\ell^{\infty}(S)\) and its relation with the left reversibility of \(S\). It is shown that if \(\ell^{\infty}(S)\) has a left invariant submean \(\mu\), and \(S=\{T_s\mid s\in S\}\) is a representation of \(S\) as a semigroup of nonexpansive mappings on a weakly compact convex subset \(C\) with normal structure in a nontrivial Banach space \(E\), then the set \(\{z\in C\mid \mu_t\| T_tx-z\| =\inf\{\mu_t\| T_tx-y\| \mid y\in C\}\}\) is a proper subset of \(C\).NEWLINENEWLINEA remarkable consequence of this result is an improvement of a fixed point theorem of Lim for left reversible semigroups of nonexpansive mappings. It is demonstrated that if the norm of \(E\) is Fréchet differentiable, then for each \(x\in C\), the set of all fixed points of \(S\) and the intersection of all left ideal orbits of \(x\) consists of at most one point.NEWLINENEWLINESome ergodic theorems are subsequently obtained and some related results are discussed. The chapter contains many historical comments, open problems, 107 references and is addressed to researchers and graduate students.NEWLINENEWLINEFor the entire collection see [Zbl 0970.54001].