Fixed Point and Non-Linear Ergodic Theorems for Semigroups of Non-Linear Mappings

Let S be a semigroup, ℓ∞(S) be the Banach space of bounded real valued functions on S with the supremum norm. There is a strong relation between the existence of an invariant mean (or submean) on an invariant subspace of ℓ∞(S) and fixed point or ergodic properties of S when S is represented as a semigroup of nonexpansive mappings on a closed convex subset of a Banach space. It is the purpose of this chapter of the Handbook to exhibit on some recent results on such relations. Since this handbook is intended for researchers and graduate students, detailed proofs for central results, historical remarks, open problems and many references will be included. It is our hope that our effort will generate further research in this direction of non-linear analysis which depends on the ideal theory of S,and existence of an invariant mean (or submean) on a subspace of ℓ∞(S) of a semigroup S.