Nonlinear ergodic theorems for almost nonexpansive curves over commutative semigroups

Let S be a commutative semigroup with identity, and let H be a real Hilbert space with inner product 〈 · , · 〉 and norm ‖ · ‖. We also denote by Z, Z, R and R the sets of all integers, nonnegative integers, real numbers and nonnegative real numbers, respectively. Let C be a subset of H. Then a mapping T : C → C is called nonexpansive if ‖Tx−Ty‖ ≤ ‖x−y‖ for all x, y ∈ C. The first nonlinear ergodic theorem for nonexpansive mappings (in a Hilbert space) was established by Baillon [1]: Let C be a nonempty closed convex subset of H and let T be a nonexpansive mapping of C into itself. If T has a fixed point, then the Cesàro means (1/n) ∑n−1 k=0 T x converge weakly as n →∞ to a fixed point y of T. In this case, put y = Px for each x ∈ C. Then P is a nonexpansive retraction of C onto the set Fix(T ) of fixed points of T such that PT = TP = P for all n ∈ Z, and Px ∈ clco{Tx : n ∈ Z} for each x ∈ C, where clco A is the closure of the convex hull of A. In [33, 34], Takahashi proved the existence of such an ergodic retraction for an amenable semigroup of nonexpansive mappings in a Hilbert space. And also Rodé [30] found a sequence of means on the semigroup, generalizing the Cesàro means on the positive integers, such that the corresponding sequence of mappings converges to an ergodic